Solitary wave interaction with a concentric porous cylinder system

Solitary wave interaction with a concentric porous

cylinder system

Z. Zhong, K.H. Wang*

Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4003, USA

Received 24 November 2004; accepted 11 May 2005

Available online 10 October 2005

Abstract

Theoretical investigations on solitary waves interacting with a surface-piercing concentric porous

cylinder system are presented in this paper. The outer cylinder is porous and considered thin in

thickness, while the inner cylinder is solid. Both cylinders are rigidly fixed on the bottom. Following

Isaacson’s [Isaacson, Micheal de St. Q., 1983. Solitary wave diffraction around large cylinder.

Journal of the Waterway, Port, Coastal and Ocean Engineering 109(1), 121–127.] approach, we

obtained the solutions for free-surface elevation and the corresponding velocity potential in terms of

Fourier integrals. Numerical results are presented to show the effects of incident wave condition,

porosity of the outer cylinder and radius ratio on wave forces and wave elevations around the inner

and outer cylinders.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Porous cylinder; Hydrodynamic force; Fourier integral

1. Introduction

Interactions of water waves with a solid vertical cylinder or cylinder arrays have been

an active research topic for years. MacCamy and Fuchs (1954) carried out one of the first

analytical studies on wave diffraction by a circular cylinder. As regarding shallow water

wave diffraction around a vertical cylinder, Isaacson (1977, 1978) was among the pioneers

to derive analytical solutions. For cnoidal waves, he expressed the incident velocity

potential in terms of a Fourier series. Then the governing equation was solved to find

Ocean Engineering 33 (2006) 927–949

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doi:10.1016/j.oceaneng.2005.05.013

* Corresponding author. Tel.: C1 713 743 4277; fax: C1 713 743 4260.

E-mail address: [email protected] (K.H. Wang).

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Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949928

the analytical solutions of scattered velocity potential. He found that both forces acting on

the cylinder and maximum run-ups around the cylinder due to cnoidal waves are

appreciably larger than the predictions of shallow water sinusoidal wave theory and closer

to experimental results. Later, using a Fourier integral transform for a first approximation,

Isaacson (1983) extended his work to the case of solitary waves interacting with a solid

vertical cylinder. Formulas for calculating forces on the cylinder and run-ups were also

obtained for this case. Basmat and Ziegler (1998) revisited this topic by considering the

second order approximation of the solitary wave. By means of a Fourier transformation

technique as used by Isaacson (1983), they attempted to develop the analytical solutions

up to the second order for the diffraction of a solitary wave by a rigid vertical cylinder.

However, to our knowledge, their solutions are not quite valid as a result of possible mis-

implementation of the second-order free-surface boundary conditions.

On the other hand, many researchers have also sought to solve the problem of shallow

water waves interacting with a vertical cylinder or cylinder arrays experimentally and

numerically. Yates and Wang (1994) conducted an experimental study of solitary wave

scattering by a vertical cylinder. Experimental data was presented for the wave elevations

and the forces on the vertical circular cylinder encountered by a solitary wave. The

nonlinear process was examined and discussed. In a series study, Wang and his research

group (Wang et al., 1992; Wang and Jiang, 1994; Jiang and Wang, 1995; Wang and Ren

1994, 1999) carried out a systematic numerical analysis of solitary and cnoidal waves

interacting with a vertical cylinder or cylinder arrays. Through these authors’ work, a

better understanding and vision of the shallow water wave diffraction by cylinder(s) are

achieved. Ohyama (1991) and Yang and Ertekin (1992) calculated the solitary wave forces

by a boundary element method. Using finite difference method, Neill and Ertekin (1997)

studied the similar problem by employing both Green–Naghdi and Boussinesq equations.

They investigated the nonlinear effects on the solitary wave forces by comparing their

results with the first approximation given by Isaacson (1983) and the numerical results

given by Wang et al. (1992). They reported that the first approximation approach for

overturning moment is underestimated remarkably when the incident wave height is

relatively large, while the agreements between their numerical results and those from the

first approximation are fairly good for the horizontal force. Their results for horizontal

force, however, are much smaller than those presented by Wang et al. (1992).

However, limited attentions have been paid to wave diffraction by a concentric two-

cylinder system, with an outer cylinder being porous, whereas an inner one being

impermeable. Among the rare ones, Wang and Ren (1994) derived a diffraction theory for

sinusoidal waves interacting with an aforementioned system. The effects of various wave

parameters and structural porosity were examined. It was shown that the hydrodynamic

force on the inner cylinder increases as the annular spacing decreases. As the annular

spacing becomes smaller, the long waves demonstrate larger forces on the inner cylinder

than do the short waves. Due to the existence of the outer porous cylinder, both

hydrodynamic force acting on the interior cylinder and wave amplitude around the

windward side of the interior cylinder are reduced if compared with the results by a direct

wave impact without an outer porous cylinder. Regarding the porous effect, it was found

that the hydrodynamic force on the inner cylinder increases as the porosity of the outer

cylinder increases, while the force acting on the exterior porous cylinder decreases as

Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 929

the porous effect increases. Teng et al. (2000) and Li et al. (2003) have also reported

similar results by employing Hankel functions for the solutions of velocity potential in the

inner domain instead of Bessel functions as did by Wang and Ren (1994). In addition, they

presented the variations of force and elevation with wave frequency expressed as the

product of wave number and the radius of the exterior cylinder. Darwiche et al. (1994) and

Williams and Li (1998) extended Wang and Ren’s (1994) work to a similar two-cylinder

system case, but with the outer cylinder being porous in the vicinity of free surface and

impermeable at some distance below the water surface, and further with the inner cylinder

mounted on a storage tank. They found that with some wave characteristics the wave field

and forces inside the porous cylinder are reduced considerably. More recently, Williams

and Li (2000) presented a semi-analytical solution for water wave interactions with an

array of porous cylinders.

In the present study, following the works of Isaacson (1983) and Wang and Ren

(1994), we investigate analytically the diffraction of solitary waves by a concentric

porous two-cylinder system. The free-surface elevation and the total net hydrodynamic

forces acting on both cylinders are determined analytically. The wave induced

overturning moments are also evaluated. This study may provide useful hydrodynamic

information for the design of a coastal porous structure, for instance a protecting

structure of water intake of a power plant. Results are presented to illustrate the effects

of various wave parameters and structural porosity on this solitary wave and dual-

cylinder interaction problem. The role played by the ratio of radii of the inner and

outer cylinders is also discussed.

2. Theoretical formulation

We confine our study in this paper to solitary waves interacting with a cylindrical

breakwater system consisting of two co-axes, vertical and surface-piercing cylinders

surrounded by a fluid of constant depth h. Both cylinders are rigidly fixed on the seabed.

The wall of the outer cylinder is uniformly porous, and its wall thickness is so small,

compared with the incident wave length, that it can be neglected, while the inner cylinder

is impermeable with solid wall. The whole fluid domain of study is therefore divided into

two regions, the inner region, U2, between the inner and outer cylinders and the outer

infinite region, U1, surrounding the cylinder system. The radius of the inner cylinder is a,

while the radius of the outer cylinder is b, so that the annular spacing between the two

cylinders is the difference of their radii. Cylindrical coordinates are employed to define the

system, with r pointing outward from the axis of the cylinders and q, the angle, measured

counterclockwise from the x-axis. The z-axis, coinciding with the axis of the cylinders,

points upward, with the plane zZ0 being the still water level, and zZKh the horizontal

bottom. The free surface elevation from the undisturbed fluid level is h(r,q,t). Fig. 1

illustrates the geometry of the problem.

A right-going solitary wave with height H and speed c is considered to propagate in the

positive x-direction to encounter the cylinder system. Following Isaacson (1983), we shall

describe an alternative representation of the incident solitary wave potential. Considered

only the first approximation, the free surface elevation, h, of an incident solitary wave is

Fig. 1. Schematic diagram of a concentric two-cylinder system.


given as

hI Z H sech2

ffiffiffiffiffiffiffiffi3H

4h3

rðxKctÞ

" #; (1)

where Cartesian coordinate system is used for the time being; the subscript, I, denotes

incident wave; the wave speed is given as cZffiffiffiffiffigh

pto the first approximation. g is the

gravitational acceleration. hI may be expressed in terms of a Fourier integral as

hI ZH

2p

ðNKN

AðkÞeikðxKctÞ dk; (2)

and the corresponding expression for the incident velocity potential, FI, is

FI ZH

2pffiffiffihg

q ðNKN

AðkÞ

ikeikðxKctÞ dk; (3)

where k denotes the wave number. The real parts of these and subsequent expressions

correspond to the physical quantities concerned. The Fourier transform A(k) of hI is

AðkÞ Z4p d3k

3Hcosech pk

ffiffiffiffiffiffiffih3

3H

r" #; (4)

The integral in Eq. (3) may be rewritten as the sum of two integrals, which gives

FI ZH

2pffiffiffihg

q ðN0

AðkÞ

ikeikðxKctÞ dkK

ðN0

AðkÞ

ikeKikðxKctÞ dk

24

35: (5)


Using the identity of

expðGikxÞ ZXN

mZ0

3m eGðipm=2ÞJmðkrÞcosðmqÞ; (6)

where Jm(kr) is the Bessel function of the first kind of order, m; 30Z1 and 3mZ2 for mR1,

the incident velocity potential can be expressed in terms of cylindrical coordinate system

as

FI ZH

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3mJmðkrÞcosðmqÞðeKiðkctKðpm=2ÞÞKeiðkctKðpm=2ÞÞÞ

dk: (7)

The fluid is assumed to be incompressible and inviscid and its motion is irrotational.

Thus, the velocity potentials satisfy the Laplace’s equation. For a monochromatic incident

wave with a frequency kc, we may separate the time factors exp(Gikct) out from the

complete velocity potential as

Fðr; q; z; tÞ Z fðr; q; zÞeHikct: (8)

If we take the potentials to be accurate to the order O[(kh)2], according to Isaacson

(1977), the complex spatial components of the velocity potentials f are independent of

vertical coordinate z and satisfy the Laplace’s equation in the horizontal plane. In

cylindrical coordinates, it reads

v2fi

vr2C

1

r

vfi

vrC

1

r2

v2fi

vq2Z 0; i Z 1; 2; (9)

where the subscript 1 refers to the fluid region, U1(rRb), and 2 refers to the region,

U2(a%r%b). Note that the usual boundary conditions, requested for the potential flow

with free surface, i.e. the bottom boundary condition and the kinematic and dynamic free

surface boundary conditions, have already been satisfied in the procedure of derivation of

Eq. (9) (Isaacson, 1977). However, f1 and f2 are still subject to the following boundary

conditions in U1 and/or U2.

(1) The boundary condition on the interior solid cylinder surface requires that the normal

velocity vanish over there

vf2

vrZ 0 on r Z a: (10)

(2) The continuity condition of fluid velocity on the porous cylinder surface

vf1

vrZ

vf2

vrZKwðqÞ on r Z b; (11)

where w(q) is the spatial component of the normal velocity W(q,t) of the fluid passing

through the porous cylinder from region U1 to region U2 and Wðq; tÞZwðqÞeHikct.


(3) The far field radiation boundary condition for the scattered wave potential

limr/N

ffiffir

p vfS

vrKikfS

� �Z 0; (12)

where fS is the spatial component of the scattered wave potential Fs(r,q,t) and

FSðr; q; tÞZfSðr; qÞeHikct.

We confine our analysis to a porous cylinder with fine pores. The fluid flow passing

through the porous cylinder can be assumed to obey Darcy’s law. Hence, the porous flow

velocity W(q,t) is linearly proportional to the pressure difference between the two sides of

the porous cylinder (Chwang, 1983; Wang and Ren, 1994). We have

Wðq; tÞ Zd

mðp1 Kp2Þ on r Z b; (13)

where m is the constant coefficient of dynamic viscosity and d is a material constant having

the dimension of a length. The hydrodynamic pressures pi(r,q,t) (iZ1,2) are related to the

velocity potentials through the linearized Bernoulli equation

pi ZKrvFi

vt; i Z 1; 2; (14)

where r denotes the constant fluid density. From Eqs. (8), (13) and (14), we have the

expression for w(q) as

wðqÞ ZGd

mrikc½f1ðb; qÞKf2ðb; qÞ�: (15)

3. Analytical solutions

For the outer region, U1, the velocity potential f1 in the presence of a dual-cylinder

system is usually expressed as the sum of the incident potential fI and a scattered potential

fS as

f11 Z f1

I Cf1S; (16)

f21 Z f2

I Cf2S; (17)

where the superscripts 1 and 2 refer to the terms with factors eKikct and eikct, respectively.

From Eq. (7), we have

f1I Z

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eiðpm=2ÞJmðkrÞcosðmqÞ

dk; (18)


f2I ZK

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eKiðpm=2ÞJmðkrÞcosðmqÞ

dk: (19)

The scattered wave potentials, f1S and f2

S, may be taken as independent of z to the first

approximation and must then satisfy the governing Eq. (9). Corresponding to the far field

radiation boundary condition, Eq. (12), f1S and f2

S can be found as

f1S Z

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eiðpm=2Þ A1

mHð1Þm ðkrÞcosðmqÞ

dk; (20)

f2S ZK

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eKiðpm=2ÞA2

mHð2Þm ðkrÞcosðmqÞ

dk; (21)

where A1m and A2

m are unknown complex coefficients; Hð1Þm and Hð2Þ

m are Hankel functions of

the first and second kind, respectively, of order m.

Similarly, the solution of Eq. (9) in the inner region U2 can be obtained as

f12 Z

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eiðpm=2Þ½B1

mJmðkrÞCC1mYmðkrÞ�cosðmqÞ

dk; (22)

f22 ZK

H

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m eKiðpm=2Þ½B2

mJmðkrÞCC2mYmðkrÞ�cosðmqÞ

dk; (23)

where B1m, C1

m, B2m and C2

m are unknown complex coefficients; Ym(kr) is the Bessel function

of the second kind of order m.

Substituting Eqs. (15)–(23) into the structural boundary conditions (10) and (11) gives

B1mJ 0

mðkaÞCC1mY 0

mðkaÞZ0;

A1mHð1Þ0

m ðkbÞKB1mJ 0

mðkbÞKC1mY 0

mðkbÞZKJ 0mðkbÞ;

A1miG0Hð1Þ

m ðkbÞKB1m½iG0JmðkbÞKJ 0

mðkbÞ�KC1m½iG0YmðkbÞKY 0

mðkbÞ�ZKiG0JmðkbÞ;

8>><>>:

(24)

B2mJ 0

mðkaÞCC2mY 0

mðkaÞZ0;

A2mHð2Þ0

m ðkbÞKB2mJ 0

mðkbÞKC2mY 0

mðkbÞZKJ 0mðkbÞ;

A2miG0Hð2Þ

m ðkbÞKB2m½iG0JmðkbÞKJ 0

mðkbÞ�KC2m½iG0YmðkbÞKY 0

mðkbÞ�ZKiG0JmðkbÞ;

8>><>>:

(25)

where the porous effect is defined as G0Z(rcd/m) (Chwang and Li, 1983). Solving Eqs. (24)

and (25), the explicit expressions of the complex coefficients A1m, B1

m, C1m, A2

m, B2m, and C2

m


are obtained as

A1mZK

J 0mðkbÞSmCi2G0

pkbJ 0

mðkaÞ

Hð1Þ0m ðkbÞSmCi2G0

pkbHð1Þ0

m ðkaÞ; (26)

B1mZK

2G0

pkbY 0

mðkaÞ


pkbHð1Þ0

m ðkaÞ; (27)

C1mZ

2G0

pkbJ 0

mðkaÞ


pkbHð1Þ0

m ðkaÞ; (28)

A2mZK

J 0mðkbÞSmCi2G0

pkbJ 0

mðkaÞ


pkbHð2Þ0

m ðkaÞ; (29)

B2mZ

2G0

pkbY 0

mðkaÞ


pkbHð2Þ0

m ðkaÞ; (30)

C2mZK

2G0

pkbJ 0

mðkaÞ


pkbHð2Þ0

m ðkaÞ; (31)

where SmZJ 0mðkaÞY 0

mðkbÞKJ 0mðkbÞY 0

mðkaÞ, and the primes of the Bessel functions indicate

the derivatives.

There exist two limiting cases. The first case is the one when aZb and G0Z0, which

means the cylinder system is simplified to a single solid cylinder. This case coincides with

Isaacson’s (1983) study case. For this limiting case, the complex coefficients are simplified

to be

A1m ZK

J 0mðkbÞ

Hð1Þ0m ðkbÞ

; (32)

A2m ZK

J 0mðkbÞ

Hð2Þ0m ðkbÞ

; (33)

B1m Z B2

m Z C1m Z C2

m Z 0: (34)

The other limiting case is solitary wave interacting with a hollow porous cylinder (aZ0), for which the complex coefficients are derived separately as

A1m ZK

½J 0mðkbÞ�2

Hð1Þ0m ðkbÞJ 0

mðkbÞC 2G0

pkb

; (35)

B1m Z

2G0

pkb

Hð1Þ0m ðkbÞJ 0

mðkbÞC 2G0

pkb

; (36)


C1m Z 0; (37)

A2m ZK

½J 0mðkbÞ�2

Hð2Þ0m ðkbÞJ 0

mðkbÞK2G0

pkb

; (38)

B2m ZK

2G0

pkb

Hð2Þ0m ðkbÞJ 0

mðkbÞK2G0

pkb

; (39)

C2m Z 0: (40)

Once the constant coefficients A1m, B1

m, C1m, A2

m, B2m, and C2

m are determined, and after

recovery of time factors, the total velocity potentials for both regions can be expressed as

F1 Z F1I CF1

S CF2I CF2

S

ZH

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m½JmðkrÞCA1

mHð1Þm ðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ

dk

KH

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m½JmðkrÞCA2

mHð2Þm ðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ

dk; (41)

F2ZF12CF2

2ZH

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m½B

1mJmðkrÞCC1

mYmðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ dk

KH

2pffiffiffihg

q ðN0

AðkÞ

ik

XN

mZ0

�3m½B

2mJmðkrÞCC2

mYmðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ dk:

(42)

Based on the derived velocity potentials, various quantities of engineering interest may

now be determined. The wave profiles in both regions are obtained from the linearized

dynamic free-surface boundary condition. Thus

h1ZK1

g

vF1

vtZ

H

2p

ðN0

AðkÞXN

mZ0

�3m½JmðkrÞCA1

mHð1Þm ðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ

dk

CH

2p

ðN0

AðkÞXN

mZ0

�3m½JmðkrÞCA2

mHð2Þm ðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ

dk; (43)


h2ZK1

g

vF2

vtZ

H

2p

ðN0

AðkÞXN

mZ0

�3m½B

1mJmðkrÞCC1

mYmðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ dk

CH

2p

ðN0

AðkÞXN

mZ0

�3m½B

2mJmðkrÞCC2

mYmðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ dk: ð44Þ

The wave run-up RI(q) around the interior cylinder is the maximum value of h2 at rZa,

whereas the wave run-up RO(q) around the exterior cylinder is the maximum value of h1 at rZb.

The total hydrodynamic forces on the inner cylinder (FIx) and the outer porous cylinder

(FOx) in the direction of wave propagation may be obtained by integrating the pressure

distributions on the cylinders with respect to z and q at rZa and b, respectively. From

linearized Bernoulli equation, the pressures acting on the peripheries of the outer and inner

cylinders are given, respectively, as

p1 ZKrvF1

vtZ

rgH

2p

ðN0

AðkÞXN

mZ0

½3m½JmðkrÞCA1mHð1Þ

m ðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ�dk

CrgH

2p

ðN0

AðkÞXN

mZ0

½3m½JmðkrÞCA2mHð2Þ

m ðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ�dk; ð45Þ

p2 ZKrvF2

vtZ

rgH

2p

ðN0

AðkÞXN

mZ0

½3m½B1mJmðkrÞCC1

mYmðkrÞ�cosðmqÞeKiðkctKðpm=2ÞÞ�dk

CrgH

2p

ðN0

AðkÞXN

mZ0

½3m½B2mJmðkrÞCC2

mYmðkrÞ�cosðmqÞeiðkctKðpm=2ÞÞ�dk: ð46Þ

Accordingly

FIx Z

ð2p

0

ð0Kh

½p2�rZacosðpKqÞa dqdz

ZKirgHha

ðN0

AðkÞ½B11J1ðkaÞCC1

1Y1ðkaÞ�eKikct dk

KirgHha

ðN0


1Y1ðkaÞ�eikct dk; ð47Þ


FOx Z

ð2p

0

ð0Kh

½p1Kp2�rZb cosðpKqÞb dqdz

ZKirgHhb

ðN0

AðkÞ½ð1KB11ÞJ1ðkbÞCA1

1Hð1Þ1 ðkbÞKC1

1Y1ðkbÞ�eKikct dk

KirgHhb

ðN0


1Hð2Þ1 ðkbÞKC2

1Y1ðkbÞ�eikct dk: (48)

The overturning moment on the interior cylinder about the sea bed can be obtained as

MI Z

ð2p

0

ð0Kh

½p2�rZa cosðpKqÞaðz ChÞ dqdz

ZKirgHh2a

2

ðN0


1Y1ðkaÞ�eKikct dk

KirgHh2a

2

ðN0


1Y1ðkaÞ�eikct dk; ð49Þ

and the overturning moment on the exterior cylinder about the sea bed can be determined

as

MO Z

ð2p

0

ð0Kh

½p1Kp2�rZbcosðpKqÞbðz ChÞdqdz

ZKirgHh2b

2

ðN0


1Hð1Þ1 ðkbÞKC1

1Y1ðkbÞ�eKikct dk

KirgHh2b

2

ðN0


1Hð2Þ1 ðkbÞKC2

1Y1ðkbÞ�eikct dk: (50)

4. Results and comments

To implement the derived analytical solutions and examine the effects of different

parameters on the scattering of solitary waves by a dual-cylinder system, a numerical

program is developed. Let us consider the case of an incident wave height HZ1.0 m. The

radius of the outer porous cylinder is chosen as 25.0 m. In addition to the porosity G0 of


the outer cylinder, the ratio of the radii of both cylinders, a/b, is used to examine the effect

of the annular spacing between the cylinders. The wave-effect parameter

Cw ZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðHb2Þ=ðh3Þ

p, which is found to be the one governing the similarity of solitary

waves interacting with the two-cylinder system as in agreement with the discovery made

by Isaacson (1983) for the case of solitary wave diffraction around a solid cylinder, is also

considered.

From the analytical formulae of forces and overturning moments on the cylinders, we

note that the values of wave-induced dimensionless overturning moments (MI/rgah2H;

M0/rgah2H) on both the inner and outer cylinders are exactly half of those of the

corresponding dimensionless forces (FIx/rgahH; F0x/rgbhH). So, the overturning

moments on the cylinders must follow the same laws as forces. Here in, we then just

show the results of hydrodynamic forces on the cylinders.

4.1. Effect of the radius ratio a/b

Figs. 2 and 3 show the maximum dimensionless hydrodynamic forces (FIx/rgahH; F0x/

rgbhH) on the interior and exterior cylinders vs. a/b at G0Z1.0 for different wave

parameters Cw. It can be seen that the hydrodynamic force on the interior cylinder

increases with an increase of a/b (or decrease of annular spacing), while the force on the

exterior cylinder shows the decreasing trend. The wave force at a/bZ1 represents the

force of solitary waves acting on an impermeable vertical cylinder and the results agree

exactly with the Isaacson’s (1983) results. Generally, waves with larger Cw exert larger

dimensionless forces on the inner cylinder, however, in a reversal trend, the forces on the

outer cylinder is smaller, typically when a/bO0.6. The hydrodynamic force acting on a

Fig. 2. Hydrodynamic forces on the interior cylinder vs. a/b for different Cw at G0Z1.0.

Fig. 3. Hydrodynamic forces on the exterior cylinder vs. a/b for different Cw at G0Z1.0.


hollow porous cylinder is determined by setting aZ0. In this limiting case, it is found that

the force on a single porous cylinder is reduced dramatically when compared with the

force acting on an impermeable cylinder of the same radius. For an incident wave of CwZ0.6, the dimensionless force acting on an impermeable cylinder is 2.0 (Fig. 2, a/bZ1).

However, the force on a porous cylinder is reduced to 0.736 as shown in Fig. 3, at a/bZ0.

Fig. 4. Hydrodynamic forces on the interior cylinder vs. a/b for different G0 at CwZ0.8.

Fig. 5. Hydrodynamic forces on the exterior cylinder vs. a/b for different G0 at CwZ0.8.


Figs. 4 and 5 show the maximum dimensionless hydrodynamic forces on the interior

and exterior cylinders vs. a/b, respectively, at CwZ0.8 for different porosity parameter G0.

For a specified G0, again, the force acting on the interior cylinder increases with an

increase of a/b, and the force on the exterior cylinder decreases. For the case of an

impermeable single cylinder, in which a/bZ1, the forces on the cylinder for different G0

approach to the same value, 2.3, as the effect of G0 no longer exists. The outer porous

cylinder can be viewed as a cylindrical breakwater to protect the inner cylinder from

experience of large wave forces. From Fig. 4, we notice that with an increase of G0, the

dimensionless forces acting on the inner cylinder increase. However, when G0O1.0, there

is no significant change in force variation. Overall, the force on an inner cylinder with an

outer porous cylinder is smaller than that on a cylinder with the same radius but without an

outer porous cylinder. For the solitary wave induced force on the outer cylinder, the results

in Fig. 5 show that with an increase of G0, the force decreases. A G0 within the range of 0.5

and 1.0 can be selected to reduce forces on both inner and outer cylinders.

Variations of maximum run-up around the inner and outer cylinders at G0Z1.0 and

CwZ0.8 for different a/b are illustrated in Figs. 6 and 7, respectively. It is clearly

shown that in the windward side the wave run-ups increase as the radii ratio a/b

increases for both interior and exterior cylinders, and in the leeward side, the wave

run-ups increase with the radii ratio a/b too for the interior cylinder, while they

decrease with a/b between 0 and p/2 for the exterior cylinder. For both interior and

exterior cylinders, somewhere in the region of [0, p/2], wave run-up reaches a

minimum. Then toward both windward and leeward sides of the cylinder it increases.

From Fig. 7, it is noted that there is almost no variation of wave run-ups with respect

to a/b in the vicinity of qZp/2 and 0 for the exterior cylinder.

Fig. 6. Maximum run-ups around the inner cylinder for different a/b at G0Z1.0 and CwZ0.8.

Fig. 7. Maximum run-ups around the outer cylinder for different a/b at G0Z1.0 and CwZ0.8.


Fig. 8. Hydrodynamic force on the interior cylinder vs. G0 for different Cw at a/bZ0.5.


4.2. Effect of the porous parameter G0

The hydrodynamic forces on the cylinders vs. G0 are shown in Figs. 8 and 9,

respectively, at a/bZ0.5 for different wave parameter Cw. As described in the previous

section, the results in Fig. 8 again reveal that the force on the inner cylinder increases as

the porosity of the outer cylinder increases for any given incident wave. However, the

wave force on the exterior porous cylinder decreases as the porous effect increases. The

dominant role played by the porous effect in the range of large G0 values is shown in

Fig. 9. The results further demonstrate that, with the existence of the exterior porous

cylinder, the hydrodynamic force acting on the interior cylinder is reduced, especially

when G0 is less than 1.0, if compared with the force exerted on the interior cylinder by a

direct wave impact. For instance, for a given incident wave and water depth, the

dimensionless hydrodynamic force acting on a cylinder of radius 12.5 m by a direct wave

impact is 1.51, whereas, with the G0Z0.5, it is reduced to 1.22 (Fig. 8, CwZ0.8) on a

cylinder of same radius with an outer porous cylinder of radius 25.0 m, a 19.2% reduction.

4.3. Effect of wave parameter Cw

As the current analytical model is developed based on a first approximation governing

equation, it generally limits to the application for incident wave with small Cw, saying

within the range of 0.0–2.0. This implies small wave-height and the radii of the cylinders

are either comparable with or smaller than the water depth. Fig. 10 illustrates the

dimensionless hydrodynamic force on the inner cylinder vs. the wave parameter Cw at a/

bZ0.5 for different values of G0. From Fig. 10, we notice that the dimensionless wave

Fig. 9. Hydrodynamic force on the exterior cylinder vs. G0 for different Cw at a/bZ0.5.


force acting on the inner cylinder increases with an increase of wave parameter Cw for

G0R1.0; however, for G0!1.0 it increases initially as the wave parameter Cw increases

until the force reaches a maximum value. The force then decreases gradually with further

increase of the wave parameter. For the case of a/bZ0.5, the forces acting on the exterior

porous cylinder vs. the wave parameter for different values of G0 are presented in Fig. 11.

Fig. 10. Hydrodynamic force on the interior cylinder vs. Cw for different G0 at a/bZ0.5.

Fig. 11. Hydrodynamic force on the exterior cylinder vs. Cw for different G0 at a/bZ0.5.


We note that the dimensionless force tends to increase initially then decreases as Cw

increases. However, with further increase of Cw, the force on the exterior porous cylinder

increases, especially for the cases when G0%1.0.

Figs. 12 and 13 show variations of maximum run-up around the inner and outer

cylinders at a/bZ0.5 and G0Z1.0 for different Cw. For both inner and outer cylinders, in

the windward side, the wave run-ups increase as the wave parameter Cw increases, but in

the leeward side of the cylinders, wave run-ups decrease as Cw increases. It is interesting to

note that the maximum run-ups around the exterior porous cylinder occur at about qZ1308

instead of the windward side of the cylinder for the cases when CwO0.6, due to the effect

of porosity of the outer cylinder.

4.4. Free-surface profiles

Let the right-going waves start from some location left of the cylinder system. The free-

surface profiles along the center line of the cylinder system at tZ0, 2, 5, and 11 for

different G0 are presented in Figs. 14–17, respectively. The profile at tZ0 illustrates the

free surface elevation when the crest of the solitary wave is approaching to the cylinder

system; at tZ2, the profile presents the moment that the wave crest is just encountering the

outer porous cylinder; tZ5 is the time when the wave crest is located at the middle of the

cylinder system; while at tZ11, the wave crest has just passed the cylinder system. By

reviewing these plots, the effect of the porosity of the porous cylinder on wave elevation

can be clearly identified. There is a reduction of surface elevation in the annular region due

to the energy dissipation by the porous cylinder, and relatively the wave amplitude in the

annular region increases as G0 increases. However, when G0 is greater than 1.0,

Fig. 12. Maximum run-ups around the inner cylinder for different Cw at G0Z1.0 and a/bZ0.5.

Fig. 13. Maximum run-ups around the inner cylinder for different Cw at G0Z1.0 and a/bZ0.5.


Fig. 14. Free-surface profile for different G0 at a/bZ0.5 and CwZ0.8 (tZ0).





the reduction of the water level is then not so significant. Meanwhile, along the leeward

side of the annular region, reduction of the wave amplitude is also found. On the other

hand, although the outer cylinder is porous, the interesting effect of the interaction

between the solitary wave and the cylinder system can still be noticed. At tZ2, the

wave run-ups reach the maximum in the windward side of the dual-cylinder system.



Within the annular spacing, the surface elevation is slightly greater than 1.0 m except for

the case of G0Z0.2, in which the surface elevation drops significantly when compared

with those of other cases as the results of large dissipation of wave energy by the porous

cylinder. It is found that the profiles at tZ5 are not symmetric about the middle of the

cylinders due to the effects of wave scattering and wave energy dissipation around the

cylinder system. At tZ11, the obvious back-scattered waves propagating outwards in the

windward side of the cylinder system can be clearly noticed.

5. Conclusions

The solitary wave interactions with a co-axes two-cylinder system consisting of an

exterior porous cylinder and an interior solid cylinder are investigated analytically. The

free surface elevation and hydrodynamic forces are determined. It is found that the

hydrodynamic force on the inner cylinder increases as the annular spacing decreases,

while the force on the exterior cylinder decreases as the annular spacing decreases. The

force on a single porous cylinder is reduced dramatically when compared with the force

acting on an impermeable cylinder of the same radius. Regarding the porous effect, we

note that the larger the porosity of the outer cylinder, the larger the hydrodynamic force on

the inner cylinder, and the smaller the force on the outer cylinder. The results also indicate

that with the existence of the exterior porous cylinder the hydrodynamic force acting on

the interior cylinder by a direct wave impact is reduced. Generally, the hydrodynamic

force increases with an increase of the wave parameter. However, the force on the outer

cylinder varies slightly differently with initial increase followed by a decreasing trend then

further increases as wave parameter increases. The results of free-surface plots show the

evidence of the reduction of the wave amplitude around the windward side of the inner

cylinder and the leeward side of the outer cylinder.

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Solitary wave interaction with a concentric porous cylinder system