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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0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
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).
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.
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949930
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)
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 931
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.
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949932
(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)
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 933
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
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949934
are obtained as
A1mZK
J 0mðkbÞSmCi2G0
pkbJ 0
mðkaÞ
Hð1Þ0m ðkbÞSmCi2G0
pkbHð1Þ0
m ðkaÞ; (26)
B1mZK
2G0
pkbY 0
mðkaÞ
Hð1Þ0m ðkbÞSmCi2G0
pkbHð1Þ0
m ðkaÞ; (27)
C1mZ
2G0
pkbJ 0
mðkaÞ
Hð1Þ0m ðkbÞSmCi2G0
pkbHð1Þ0
m ðkaÞ; (28)
A2mZK
J 0mðkbÞSmCi2G0
pkbJ 0
mðkaÞ
Hð2Þ0m ðkbÞSmCi2G0
pkbHð2Þ0
m ðkaÞ; (29)
B2mZ
2G0
pkbY 0
mðkaÞ
Hð2Þ0m ðkbÞSmCi2G0
pkbHð2Þ0
m ðkaÞ; (30)
C2mZK
2G0
pkbJ 0
mðkaÞ
Hð2Þ0m ðkbÞSmCi2G0
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)
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 935
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)
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949936
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
AðkÞ½B21J1ðkaÞCC2
1Y1ðkaÞ�eikct dk; ð47Þ
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 937
FOx Z
ð2p
0
ð0Kh
½p1Kp2�rZb cosðpKqÞb dqdz
ZKirgHhb
ðN0
AðkÞ½ð1KB11ÞJ1ðkbÞCA1
1Hð1Þ1 ðkbÞKC1
1Y1ðkbÞ�eKikct dk
KirgHhb
ðN0
AðkÞ½ð1KB21ÞJ1ðkbÞCA2
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
AðkÞ½B11J1ðkaÞCC1
1Y1ðkaÞ�eKikct dk
KirgHh2a
2
ðN0
AðkÞ½B21J1ðkaÞCC2
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
AðkÞ½ð1KB11ÞJ1ðkbÞCA1
1Hð1Þ1 ðkbÞKC1
1Y1ðkbÞ�eKikct dk
KirgHh2b
2
ðN0
AðkÞ½ð1KB21ÞJ1ðkbÞCA2
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
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949938
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.
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 939
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.
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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.
Z. Zhong, K.H. Wang / Ocean Engineering 33 (2006) 927–949 941
Fig. 8. Hydrodynamic force on the interior cylinder vs. G0 for different Cw at a/bZ0.5.
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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.
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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.
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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.
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Fig. 14. Free-surface profile for different G0 at a/bZ0.5 and CwZ0.8 (tZ0).
Fig. 15. Free-surface profile for different G0 at a/bZ0.5 and CwZ0.8 (tZ2).
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Fig. 16. Free-surface profile for different G0 at a/bZ0.5 and CwZ0.8 (tZ5).
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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.
Fig. 17. Free-surface profile for different G0 at a/bZ0.5 and CwZ0.8 (tZ11).
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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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