Hydraulic performance of vertical walls with horizontal slots used as breakwater

Coastal Engineering 57 (2010) 745–756

Contents lists available at ScienceDirect

Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g

Hydraulic performance of vertical walls with horizontal slots used as breakwater O.S. Rageh a, A.S. Koraim b,⁎ a b

Irrigation and Hydraulics Department, Faculty of Engineering, El-Mansoura University, El-Mansoura, Egypt Water and Water Structure Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt

a r t i c l e

i n f o

Article history: Received 15 March 2009 Received in revised form 1 March 2010 Accepted 23 March 2010 Available online 15 May 2010 Keywords: Vertical breakwaters Slotted Transmission Reflection Energy dissipation

a b s t r a c t The hydrodynamic performance of a vertical wall with permeable lower part (horizontal slots) was experimentally and theoretically studied under normal regular waves. The effect of different wave and structural parameters was investigated e.g. the wave length, the upper part draft, and the lower part porosity. Also, the theoretical model based on an Eigen Function Expansion Method and a Least Square Technique was developed. In order to examine the validity of the theoretical model, the theoretical results were compared with the present experimental results and with the results obtained from different previous studies. Comparison between experiments and predictions showed that the theoretical model provides a good estimate of the wave transmission, reflection, and energy dissipation coefficients when the friction factor f = 5.5. In general, the tested model gives transmission coefficients less than 0.5 and reflection coefficients larger than 0.5 when the relative wave length h/L is larger than 0.3, the relative upper part draft D/h larger than 0.36, and lower part porosity ε less than 0.5. Also, the tested model dissipates about 50% of the incident wave energy when the relative wave length h/L is in the range of 0.25 to 0.35. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Coasts play an important role in economy of each country for their strategic location for residential, recreational, and industrial activities. Hence, a need has arisen to protect and maintain these coasts against waves and currents. It is necessary to consider cost-effective and environment friendly structures that decrease the effect of these waves and currents before they reach the coast. In general, the width and the weight of the traditional type breakwaters (rubble mound and gravity types) increase with water depth, requiring a great amount of construction material and high sea bed bearing capacity. Also, these types block littoral drift and cause severe erosion or accretion in neighboring beaches. In addition, they prevent the circulation of water and so deteriorate the water quality near the coast. In some places, they obstruct the passage of fishes and bottom dwelling organisms. For solving the above-mentioned problems, permeable thin structures are suggested. The simplest permeable structure may be a pre cast walls which are connected to the supporting piles at the site. Each pre-cast wall is divided in to two parts; the upper part is impermeable and extends above the water level to some distance below sea level. The other part of the wall is permeable and consists of

⁎ Corresponding author. Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt. Tel.: +20 12 570 9794; fax: +20 55 230 4987. E-mail addresses: [email protected] (O.S. Rageh), [email protected] (A.S. Koraim). 0378-3839/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2010.03.005

closely spaced horizontal slots. This type helps in dissipating energy of the incident sea wave and protecting shorelines from erosion and wave attack. It also, possesses desirable features e.g. by minimizing the pollution near shore because it permits the flow exchange between the partially enclosed water body and the open sea. The present investigation with vertical wall with horizontal slots breakwater has been carried out with the following objectives: 1. To develop a simple theoretical solution for estimating the wave transmission, reflection and energy dissipation characteristics of the proposed vertical wall. 2. To investigate experimentally the same characteristics for different wave climate and structure configurations. The study of wave transmission, reflection and energy dissipation due to this structure is the key information necessary to understand the hydraulic performance of this structure as a special breakwater. The information on the characteristics of wave transmission is essential to select the appropriate configuration (void ratio, draft, etc.) for a prevailing wave climate once the permissible range of transmission is decided for the protected areas. Knowledge of wave reflection in the vicinity of this structures is necessary to select its crest level. 2. Literature review Many experimental and theoretical studies were carried out determining the efficiency of models similar to the proposed model.

746

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

These models are mainly semi-immersed smooth solid walls, slotted or screen breakwater, and bodies supported on one or more rows of widely or closely spaced piles. The available literature studies in this field assumed that no wave over topping occurred at these structures. There are many vertical permeable structures similar to the proposed vertical wall built around the world such as, (1) a steel pipe breakwater at Port of Osaka, Japan, (2) a concrete pipe breakwater at Pass Christian, USA, (3) a steel pipe breakwater at Pelangi Beach Resort, Langkawi, Malaysia, (4) a wooden pile groin at the southern coast of the Baltic Sea, German, and (5) a curtain wall–pile breakwater at the Port of Yeoho, Korea. 2.1. Semi-immersed solid breakwaters The efficiency of the semi-immersed walls was experimentally and theoretically studied by many researchers. Wiegel (1960), Reddy and Neelamanit (1992), Heikal (1997), and Koraim (2005) carried out experimental studies to determine the efficiency of this type under regular waves. Ursell (1947), Liu and Abbaspour (1982), Abul-Azm (1993), Heikal (1997), and Koraim (2005) studied this type using different theoretical models such as Power Theory, Eigen Function Expansion, Boundary Integral Equation, and Reynolds Average Navier Stokes. 2.2. Slotted or screen breakwaters Kriebel (1992), Isaacson et al. (1998), Isaacson et al. (1999), AbdelMawla and Balah (2001), Balaji and Sundar (2002), Kriebel (2004), Koraim (2005), Huang (2007) and Koutandos (2009) investigated the hydrodynamic performance of the totally and semi-immersed, single and double rows of slotted breakwaters (vertical slots) experimentally and theoretically. Also, Bergmann and Oumeraci 1998, Clauss and Habel (1999), Bergmann (2000), Koether et al. (2000), Koether (2002), Balaji and Sundar (2004), and Krishnakumar et al. (2008) investigated different types of wave screen breakwaters consisted of horizontal slots using different physical and theoretical models. The hydrodynamic performance of the suspended pipe breakwater was evaluated by Mani and Jayakumar (1995), Mani (1998), Galal (2002), and Rao et al. (2003) using physical models. 2.3. Structures supported on piles Sundar and Subbarao (2002) carried out the experimental studies on the quadrant front face pile supported breakwaters. The bottom portion consists of closely spaced piles and the top portion consists of a quadrant solid front face on the seaside. The leeward side of the top portion is a vertical face. Suh et al. (2006), (2007) studied experimentally and theoretically the hydrodynamic characteristics of a curtain–wall–pile breakwater. The upper part of this model is a vertical wall and the lower part consists of an array of vertical square and circular piles. Laju et al. (2007) studied the pile supported double skirt breakwater. The breakwater model consists of double impermeable wave barrier near the free surface extends above the water level to some distance below sea level and supported on closely spaced vertical piles. The most important literature studies related to the present study are listed in Table 1. For most studies listed, Table 1 presented some details of each study such as structure type, study objectives, theoretical model type, experimental facilities (wave flume dimensions and used water depths), wave type and main investigated wave and structure parameters. The detailed literature review reveals that there are some references available on similar models. There is no work, in particular experimental and theoretical investigations, carried out for studying the effect of the permeability of a breakwater when the slots are horizontal. Also, there is no theoretical work solving the present case

using the combination between an Eigen Function Expansion and a Least Square Methods under regular or irregular waves. The two above-mentioned reasons are the main motivations for the present work. In this paper, the wave transmission, reflection, and energy dissipation of the vertical wall with permeable lower part (horizontal slots) under regular waves were experimentally and theoretically studied. The effect of different wave and structural parameters on the hydrodynamic characteristics was investigated e.g. the wave length, the upper part draft ratio, and the lower part porosity. Also, the theoretical model based on an Eigen Function Expansion Method and a Least Square Technique was developed to study the hydrodynamic breakwater performance. 3. Theoretical model Let us consider the breakwater sketched in Fig. 1, in which h is the constant water depth in still water; D [m] is the draft of the upper part; b [m] is the breakwater thickness; w [m] is the slot width; c [m] is the clear gap between the two neighboring slots. A Cartesian coordinate system (x, z) is defined with the positive x direction toward shore from the centerline of the breakwater and the vertical coordinate z being measured vertically upwards from the still water level. A regular wave train with wave height Hi [m] is incident in the positive x-direction. A regular waves were used in the present theoretical model to simplify the solution as used in several previous works. The fluid domain is divided into two regions where region 1 (ϕ1) is located seaward the breakwater at x ≤ 0 and region 2 (ϕ2) is located shoreward the breakwater at x ≥ 0. 3.1. Velocity potential and boundary conditions The analysis is carried out assuming incompressible fluid and irrotational flow motion, the velocity potential exists, which satisfies the Laplace equation. Assuming periodic motion in time t and applying the linearized free surface boundary condition and impermeable bottom boundary condition, the velocity potential [ϕP(x,z,t), p = 1,2] may be expressed for the sea side and the shore side regions as follows, Isaacson et al. (1998) and Suh et al (2006):
 igHi 1 −iωt ϕp ðx; zÞe ϕp ðx; z; tÞ = Re − 2ω coshðkhÞ

p = 1; 2

ð1Þ

in which pffiffiffiffiffiffiffiffiRe is the real part of the complex expression between the [ ], i = −1, g [m/s2] is the acceleration of gravity, ω [1/s] is the angular wave frequency (ω = 2 π/T), T [s] is the wave period, k [1/m] is the wave number (k = 2 π/L), L [m] is the wave length, and p = 1, 2 refers to the two wave regions at the seaward and shoreward respectively. It is assumed here that the wall thickness b [m] is very small compared with the wave length L (b bb L), so that the wall has no thickness mathematically. Then ϕP(x,z) must satisfy the following matching conditions at x = 0: ∂ϕ1 ∂ϕ2 = =0 ∂x ∂x

for

0 N z N −D

∂ϕ1 ∂ϕ2 = = −iG ðϕ2 −ϕ1 Þ ∂x ∂x

for −D N z N −h :

ð2Þ

ð3Þ

The first matching condition, Eq. (2) describes that the horizontal velocities disappear on both sides of the upper impermeable part of the breakwater. The second one, Eq. (3), for the lower part of the breakwater describes that the horizontal velocities in the two regions must be same at the breakwater and that the horizontal velocity at the opening is proportional to the difference of velocity potentials, or the pressure difference, across the breakwater. The proportional constant

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

747

Table 1 Characteristics of experimental and theoretical studies of the literature review. Reference

Structure type

Ursell (1947)

Semi-immersed, thin, vertical, solid barrier Wiegel (1960) Semi-immersed, thin, vertical, solid barrier Liu and Abbsorp. 1982 Semi-immersed, thin, sloped, solid barrier Kriebel (1992) Single thin, vertical, slotted barriers Reddy and Neelamani Semi-immersed, thin, 1992 vertical, solid barrier Abul-Azm (1993) Semi-immersed, thin, vertical, solid barrier Mani and Jayakumar Single suspended pipe (1995) breakwater Heikal (1997) Semi-immersed, thin, vertical, solid barrier Isaacson et al. (1998) Semi-immersed, thin, vertical, slotted barrier Mani (1998) Single suspended pipe breakwater Clauss and Habel Under filter system (1999) Isaacson et al. (1999) Semi-immersed, vertical, slotted double barrier Bergmann (2000)

Abdel-Mawla and Balah (2001) Balaji and Sundar (2002) Galal (2002)

Koether, 2002 Sundar and Subbarao (2002) Rao et al. (2003) Koraim (2005)

Suh et al. (2006) Huang (2007) Laju et al. (2007) Suh et al. (2007) Krishnakumar et al. (2008) Koutandos 2009

Thin, vertical screen single, double and triple barriers (horizontal slots) Thin, vertical and sloped slotted double and triple barriers Double screen (double rows of horizontal pipes) Single and double suspended pipe breakwater Triple underwater filter system (horizontal slots) Quadrant front face pile supported breakwater Suspended pipe breakwater Single and double rows of vertical circular and square piles Pile-supported vertical single wall (square piles) Single and double thin, vertical, slotted barriers Pile-supported vertical double walls (circular piles) Pile-supported vertical single wall (circular piles) Partially submerged slotted wave screens Vertical Semi-Immersed Slotted Barrier

Research objectives (studied characters)

Theoretical model

kt

Deep Water Theory –

Regular

kh = 0.2–2.0 D/h = 0–0.95

kt

Power Theory

Regular

kh = 0.2–2.0 D/h = 0–0.95

kt, kr and forces

Boundary Integral Equation Empirical Equations Power Theory

10 × 0.3 × 1.2 h = 0.8 –

Regular

–

Regular

10 × 0.3 × 1.2 h = 0.8 –

Regular

kh = 0.2–2.0 D/h = 0–0.95 θ = −45°− 45° h/L N 0.5 Hi/L = 0.025–0.07 ε = 0.1–0.5 h/L = 0.8–2.5 Hi/L = 0.005–0.14 D/h = 0.06–0.5 kh = 0.2–6.0 D/h = 0–1.0 Phase

30 × 2.0 × 1.5 h = 1.0 15 × 0.3 × 0.5 h = 0.8 20 × 0.62 × 1 h = 0.45 30 × 2.0 × 1.5 h = 1.0 80 × 4.0 × 2.5 h = 1.5 20 × 0.62 × 1 h = 0.45

Regular

kt and Forces kt and kr kt and kr kt and Forces kt and H/Hi distribution kt, kr,kd, Rup and forces kt and Forces kt, kr and kd kt, kr,kd, Rup and forces

Experiment facilities Wave type [flume dim. and h in m]

Eigen Function Expansion – Eigen Function Expansion Eigen Function Expansion Empirical Equations Reynolds Average Navier Stokes Equ. Eigen Function Expansion

Regular

Regular Regular and irregular Regular Regular and irregular Regular

Main parameters ranges

Hi/gT2 = .002–.016 D/h = 0.26–0.56 ε = 0.1–0.5 ω2h/g = 0.1–2.0 D/h = 0.2–0.8 sb = 0, 1, and 2 % kh = 0.6–2.5 Hi/L = 0.02–0.09 D/h = 0.5 and 1.0 ε = 0.05–0.5 Hi/gT2 = .003–.022 D/h = 0.26–0.56 ε = 0.11–1.0 Hi/gT2 = 0.01–0.05 ds/h = 0.4–1.2 ε = 0.0–0.5 kh = 0.5–2.5 Hi/L = 0.07 D/h = 0.5 and 1.0 B/h = .22,0.55,1.1 ε = 0, 5 and 10 % h/L = 0.05–0.183 Hi = 0.5–1.5 m T = 4.5–12 s. ε = 11–40.5 %

kt, kr,kd, pressure and forces

–

307 × 5 × 7 h = 3.25–4.75 m

Regular

kt, kr and kd

–

13 × 0.3 × 0.5 h = 0.3

Regular

Hi/gT2 = .001–.016 ε = 0.3, 0.15 0.3, 0.25, 0.15 θ = 450

kt, kr,kd, and pressure kt, kr, and kd

–

Regular

–

72.5 × 2 × 2.7 h = 0.95 13 × 0.3 × 0.5 h = 0.3

B/L = 0.2–0.65 B/d = 0.53–2.11 ε = 0.238, 0.059 0.059, 0.059 Hi/gT2 = .002–.018 D/h = 0.26–0.56 ε = 0.18–0.41 θ = 450,900

kt, kr, and forces

–

kt, kr,kd, pressure and forces kd

–

kt

Eigen Function Expansion

307 × 5 × 7 h = 4.0–5.0 m 72.5 × 2 × 2.7 h = 0.8,0.9,1 50 × .71 × 1.1 h = 0.4, 0.5 15 × 0.3 × 0.5 h = 0.15,0.2,0.25

kt, kr,kd, Rup and forces kt and kr

Eigen Function Expansion Eigen Function Expansion Eigen Function Expansion Eigen Function Expansion Boundary Integral Equation

104 × 3.7 × 4.6 h = 2.4 12 × 0.3 × 0.5 h = 0.3 72.5 × 2 × 2.7 h = 1.0 104 × 3.7 × 4.6 h = 2.4 72.5 × 2 × 2.7 h = 0.95

Reynolds Average Navier Stokes Equ.

20 × 0.62 × 1 h = 0.45

kt, kr,kd and Rup kt, kr,kd, Rup and forces kt, kr,kd, and pressure kt, kr,kd, and kinematics' velocity

–

Regular

Regular and irregular Regular Regular Regular

Regular and irregular Regular Regular Regular and irregular Regular and random waves Regular

B/L = 0.25–1.25 Hi = 0.5–1.5 m T = 3–12 s. ε = 0–43 % ka = 0.6–2.0 D/h = 0.3–0.45 ε = 0.5,0.75,0.83 Hi/gT2 = .007–0.01 D/h = 0.1–0.6 ε = 0, 0.125,0.25 kh = 0.25–2.25 Hi/L = 0.005–0.06 B/d = 3, 6 ε = 0.1–0.45 kh = 0.9–4.2 Hi/L = 0.003 D/h = 0.2–0.6 ε = 0.5 kh = 1.0–2.5 Hi /L = 0.01–0.04 B/L = 0.05–0.5 ε = 0.1–0.3 kh = 1.0–4.0 B/h = 0.5, 1.0 D/h = 0.1–0.5 ε = 0.25 kh = 0.9–4.2 Hi/L = 0.003 D/ h = 0.2–0.6 ε = 0.5 D/h = 0.3, 0.5, 1.0 Hi = 0.05–0.25 m T = 1–2.5 s. ε=16.7, 9.1 % kh = 0.6–2.5 Hi/L = 0.02–0.09 D/h = 0.5 and 1.0 ε = 0.05–0.5

where a: Radius of the quadrant front face (Sundar and Subbarao (2002)); B: Space between the double rows; d: Pipe or pile diameter; ds: Height of the filter system (Clauss and Habel (1999)); sb: Seabed slope (Heikal (1997)); and θ: Breakwater inclination angle.

G\ = G/b [1/m], G [−] is the permeability parameter that is generally complex. The real part of G corresponds to the resistance of the slots and the imaginary part of G corresponds to the phase differences between the velocity and the pressure because of inertial effects. There are several ways to express the permeability parameter G. In the

present study, the method of Sollitt and Cross (1972), and Isaacson et al. (1998) is adopted and G is expressed by: G=

ε : f−is

ð4Þ

748

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

Fig. 1. Schematic diagram for the breakwater model.

In which, ε [−] is the porosity of the lower part [ε = c/(c + w)], f [−] is the friction coefficient which comes from a linearization of the velocity squared term associated with the head loss across the permeable part. In the original formulation of Sollitt and Cross (1972) f is calculated implicitly using the Lorentz principle of equivalent work so that the nonlinear effects of wave steepness are retained. In the present study the formulation of Yu (1995) is followed such that f is treated simply as a constant which is assumed to be known, and s [−] is the inertia coefficient given by

s = 1 + cm

  1−ε ε

3.2. Flow potential solution The reduced velocity potentials seaward, ϕ1, and shoreward, ϕ2, are obtained using the Eigenfunction expansion method as in Isaacson et al. (1998) and Suh et al (2006). The velocity potentials are expressed in a series of infinite number of solutions as follows: ∞

μn x

ϕ1 ðx; zÞ = ϕI − ∑ An cos μn ðz + hÞe n=0 ∞

ð5Þ

−μn x

ϕ2 ðx; zÞ = ϕI + ∑ An cos μn ðz + hÞe n=0

ð6Þ

ð7Þ

where, ϕI is an incident wave potential which is given as: in which Cm is added mass coefficient which is treated as a constant (Cm = 0) as suggested by Isaacson et al. (1998).

ikx

ϕI ðx; zÞ = cosh kðz + hÞe :

Fig. 2. Details of wave flume, model position and wave recording locations.

ð8Þ

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

749

An are unknown complex coefficients and µn for n ≥ 1 for nonpropagating evanescent waves are the positive real roots of the following dispersion relation, taken in ascending order:

where, An⁎, µn⁎, µ0⁎, and G\⁎ are the complex conjugates of An, µn, µ0 , and G\. Also,

ω2 = −gμn tan ðμn hÞ for n ≥ 1

∂GðzÞ = ∂Am = ðμm −2iG Þ cos μm ðz + hÞ

ð9Þ

∂GðzÞ = ∂Am = μm cos μm ðz + hÞ

0 N z N −h

ð19Þ

−D N z N −h:

ð20Þ

Substituting into Eq. (16) using Eq. (17) to Eq. (20), then: μ0 itself corresponds to the imaginary root of the above equation for propagating waves, (and n = 0) such that μ0 = −ik, with the wave number k being given as the real root of the corresponding equation

∞

∑ A⁎n ηnm = α0m

ð21Þ

n=0

in which 2

ω = −gk tanhðkhÞ:

ð10Þ

  h i { {

ηnm = μn⁎ μm Snm ð−D; 0Þ + μ ⁎n + 2iG ⁎ μ m −2iG Snm ð−h; −DÞ ð22Þ

* {

α0m = −μ0 μm S0m ð−D; 0Þ− μm −2iG S0m ð−h; −DÞ

3.3. Dual series relations Substituting Eq. (6) and Eq. (7) into the boundary conditions at the breakwater, Eq. (2) and Eq. (3), when x = 0, yields:

ð23Þ

where q

Snm ðp; qÞ = ∫ cos μn ðz + hÞ cos μm ðz + hÞdz

∞

∑ An μn cos μn ðz + hÞ = −μ0 cos μ0 ðz + hÞ

n=0

0 N z N −D

∞

∑ An ðμn −2iG Þ cos μn ðz + hÞ = −μ0 cos μ0 ðz + hÞ

n=0

ð11Þ

ð24Þ

p

2 3q sinðμn + μm Þðh + zÞ + sinðμn −μm Þðh + zÞ7 16 ðμn + μm Þ = 4 5 n; m ðμn −μm Þ 2

−D N z N −h:

p

ð12Þ Eqs. (11) and (12) are known as dual series relations as in Dalrymple and Martin (1990), and they are to be solved for the values of the coefficients An. The two conditions can be combined to make one mixed boundary condition which specifies the potential along the z-axis as follows: ∞

GðzÞ = ∑ An μn cos μn ðz + hÞ + μ0 cos μ0 ðz + hÞ n=0

0 N z N −D

ð13Þ

= 0; 1; 2; :: The Eq. (21) can be rewritten in the matrix form as follows: 2

η00 6 η10 6 6 ⋅ 6 4 ⋅ ηn0

η01 η11 ηn1

2 ⁎3 3 A0 η02 ………η0m 6 ⁎ 7 6 A1 7 η12 ………η1m 7 7 76 7 76 ⋅ 7= 76 6 56 7 7 4 ⋅ 5 ηn2 ………ηnm ⁎ An

3 α00 6 α01 7 7 6 6 ⋅ 7: 7 6 4 ⋅ 5 α0m 2

ð25Þ

∞

GðzÞ = ∑ An ðμn −2iG Þ cosμn ðz + hÞ + μ0 cos μ0 ðz + hÞ−D N z N −h: n=0

ð14Þ

3.4. Solution of the system of equations The Least Square Technique, suggested by Dalrymple and Martin (1990), may be used to determine the coefficients An which requires the value of: 0

j j

2

∫ GðzÞ dz = 0 −h

ð15Þ

to be minimum. Minimizing this integral with respect to each of the An coefficients leads to the following equation: 0

∫ G*ðzÞ∂GðzÞ = ∂Am dz = 0

m = 0; 1; 2; ::::::

ð16Þ

−h

where G⁎ (z) is the complex conjugate of G(z) and:

By solving Eq. (25) for evanescent wave modes, N, the equation turn to a set of N N linear simultaneous equations with unknown complex coefficients A⁎n. These equations are solved using a Fortran program based on standard matrix inversion technique (GaussElimination Technique). Then the potential functions representing the different regions, the reflection and the transmission coefficients can be determined. 3.5. Reflection, transmission and energy dissipation coefficients The theoretical reflection and transmission coefficients, denoted kr and kt, respectively are given in the first term of An coefficients (A0) by kr = j A0 j

ð26Þ

kt = j 1 + A0 j :

ð27Þ

To get a simple formula for A0, the propagating wave mode only (n = m = 0) is considered. By substituting n = m = 0 in Eq. (21), A⁎0 which is the complex conjugate of A0 can be obtained as follows: A⁎0 = α00 = η00

ð28Þ

∞

⁎ G ðzÞ = ∑ A⁎n μ ⁎n cos μ n ðz + hÞ + μ ⁎0 cos μ0 ðz + hÞ 0 N z N −D ð17Þ n=0

G⁎ ðzÞ

∞

= ∑

n=0

A⁎n



μ ⁎n

+ 2iG

{⁎



where

cosμn ðz + hÞ +

cosμ0 ðz + hÞ−D N z N −h

ð18Þ

{

{

η00 = k S00 ð−D; 0Þ + ðk + 2G ⁎ Þðk + 2G ÞS00 ð−h; −DÞ

ð29Þ

h i 2 { α00 = − k S00 ð−D; 0Þ + kðk + 2G ÞS00 ð−h; −DÞ

ð30Þ

2

μ ⁎0

750

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

El-Mansoura, Egypt. The tests were carried out to determine the hydrodynamic coefficients by the suggested vertical wall model using different wave and structural parameters. These tests were conducted to verify the results of the developed theoretical model. 4.1. Model scale Froude scaling is adopted for physical modeling, which allows for the correct reproduction of gravitational and fluid inertial forces. A scale of 1:30 is chosen for the selection of model dimensions and wave properties in the present study. The proposed breakwater can be used for water depths ranging from 10 m to 15 m. However, the present study is carried out in the laboratory for a constant water depth of 0.50 m, slots width and gap of 0.02 m, and wave period ranges from 0.9 to 1.9 s. These ranges correspond to 15 m water depth, 0.6 m slots width and gap, and 4.9 to 10.4 s wave period in the prototype respectively. 4.2. Test facility The wave flume used here was 15 m long, 1 m wide, and 1 m deep. A flap type wave generator was installed at one end of the flume and a wave absorber in the form of porous beach was installed at the other end. The experiments were carried out with a constant water depth (h) of 0.5 m and with generator motions corresponding to regular wave trains with fifteen different wave periods T = 0.9, 0.95, 0.96, 0.98, 0.99, 1.01, 1.03, 1.05, 1.07, 1.15, 1.22, 1.36, 1.53, 1.71 and 1.9 s. 4.3. Model details

Fig. 3. Variation of wave elevation with time at the different wave recording positions for the case of D = 0.18 m and T = 0.9 s.


 0 sinh2kh− sinh2kðh−DÞ 2 S00 ð−D; 0Þ = ∫ cosh kðz + hÞdz = 0:5 2k + D −D ð31Þ
 −D sinh2kðh−DÞ 2 + h−D : S00 ð−h; −DÞ = ∫ cosh kðz + hÞdz = 0:5 2k −h ð32Þ The energy equilibrium of an incident wave energy attacking the structure can be expressed as follows: Ei = Er + Et + Ed

ð33Þ

in which, Ei is the energy of incident wave, Er is the energy of the reflected wave, Et is the energy of transmitted wave and Ed is the dissipated wave energy. Based on Eq. (33), the part of the wave energy which is dissipated at the structure can be estimated as a function of the reflection and transmission coefficients as given by Isaacson et al. (1998) and Suh et al (2006): 2

2

kd = 1−kr −kt

ð34Þ

in which kd is the wave energy dissipation coefficient. 4. Experimental work A series of experiments were conducted in the Irrigation and Hydraulic Laboratory, faculty of engineering, El-Mansoura university,

The suggested breakwater model consisted of two parts. The upper part was impermeable with constant thickness (b = 0.025 m) and with variable draft (D = 0.1, 0.18, and 0.26 m). The lower part was permeable and consisted of closely spaced horizontal slots of 0.02 m width, 0.025 m thickness, and 0.02 m clear distances between slots resulting in a fixed porosity of ε = 0.5. The breakwater model was placed in the middle of the wave flume so that the distance between the centerline of the model and the flap plate or the wave absorber is 6 m each. 4.4. Wave height measurements The water level variations which resulted from the wave–structure interaction were recorded by using a Sony MVC-CD 500 Digital Still Camera. The camera zoom was adjusted exactly perpendicular to the linear scale on the glass flume side at each recording positions. The used camera was fixed on vertical stand to avoid variations of the video shots. The recording time for each run is about 2.5 the time required for a generated wave to travel from wave generator to the recording position. By using a slow motion technique (e.g. Adobe Premiere), the recorded waves taken by the camera can be analyzed and then, the wave elevation with the time can be drawn. The vertical distance between the highest and the lowest elevations represents the wave height and the distance between the two adjacent crests represents the wave period. To measure the incident (Hi) and the reflected (Hr) wave heights, two recording positions (P2 and P1) were determined in front of the breakwater model (wave generator side) at distances 0.2 L and 0.6 L respectively. This is according to the two point method of Goda and Suzuki (1976) (the distance between the two positions ranging from 0.05 to 0.45 L). To measure the transmitted wave heights (Ht), one recording position (P3) was determined behind the breakwater model (wave absorber side) at a distance of 1.5 m. Each test was done in three runs because of only one camera was used. The measured transmission and reflection coefficients are defined as the appropriate ratios of wave heights: kt = Ht/Hi and kr = Hr/Hi. The details of the

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

751

its shape. At the beginning of the formation of the partial standing wave some disturbance takes place. After that, the wave tends to be stable for some time (t = 12 to 18 s) which is the period of the suitable zone to estimate the reflection coefficients. After this time, the form of the partial standing wave changes due to the new reflection of the wave from the wave generator, which generates a new incident wave with different characteristics. Fig. 3c shows that at the first 10 s, the wave travels from the wave generator and a cross the model until it reaches the location P3. Some disturbance appears in the shape of the transmitted wave for a few seconds then, the wave seems to be very stable for some time (t = 12 to 20 s) which is the suitable period to estimate the transmission coefficients. The effect of the evanescent wave modes (N) on the kt and kr for different kh when D/h = 0.5 and ε = 0.5 is shown in Fig. 4. The figure shows that the effect of the number of the evanescent wave modes disappear after N = 5, for all investigated h/L. In addition, the effect of N decreases with h/L increasing. Throughout this study, the number of the evanescent wave modes are taken as 20 to cover the largest number of possible modes.

5.2. Verification of the theoretical model Fig. 5 presents the variation of the experimental and theoretical transmission coefficient (kt) with the relative wave length (h/L) and

Fig. 4. Effect of the evanescent wave modes (N) on the theoretical transmission and reflection coefficients for different kh when D/h = 0.5, ε = 0.5, and f = 5.5.

wave flume, the position of the tested breakwater models, and the wave recording positions are shown in Fig. 2. 5. Result analysis and discussions About 45 experimental tests were carried out to check the validity of the present theoretical model in predicting the transmission, reflection and wave energy dissipation coefficients. The analysis is presenting the performance of the proposed vertical wall in form of the relationships between transmission, reflection and energy dissipation coefficients (kt, kr and kd) and the dimensionless parameters represent the wave and structure characteristics as in the following equation: kt ; kr ;

  2 and kd = f h = L; D = h; ε; f ; Hi = gT :

ð35Þ

In general when a structure is installed in a marine environment, the presence of that structure will alter the flow pattern in its immediate neighborhoods, resulting in one or more of the following phenomena; (1) contraction of flow, (2) formation of a horseshoe vortex in front of the structure, (3) formation of lee–wake vortices (with or without vortex shedding) behind the structure, (4) generation of turbulence, (5) occurrence of reflected and diffracted waves, and (6) occurrence of wave breaking. These phenomena affect the dissipation of wave energy in addition to the dissipation caused by the structure itself (Reddy and Neelamanit 1992). 5.1. Analysis of results Fig. 3 shows a sample of the analyzed data using the slow motion technique (Adobe Premiere) at the three wave recording positions P1, P2, and P3 for the case of D = 0.18 m and T = 0.9 s. Fig. 3a and b shows that the first 10 s, the wave travels from the wave generator to the wave gauge at location P1 and P2, and reflects from the upward face of the vertical wall model and the partial standing wave begins to build

Fig. 5. Comparison between the experimental and theoretical results of the transmission coefficient for different f and D/h.

752

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

friction factor (f) for different breakwater draft ratios (D/h). The figure shows that the experimental kt decreases with increasing values of both h/L and D/h. This can be explained by considering the water particle motions. As h/L increases the water particle velocity and acceleration increases. When the wave comes across the breakwater model, the water particle velocity and acceleration suddenly change and the turbulence caused due to this sudden change in the particle motion causes the dissipation of wave energy (Rao et al., 2003). In addition, as D/h increases, the area which water path through decreases then the transmitted wave energy decreases. In more detail, kt decreases from 0.81 to 0.43, 0.78 to 0.31, 0.72 to 0.2 when D/h = 0.2, 0.36, and 0.52 respectively as h/L increases from 0.13 to 0.41. The scatter in the variation of the experimental kt with h/L is almost negligible, the coefficient of determination R2 (determined from the regression analysis) ranged from 0.97 to 0.99. Fig. 5 also shows that the theoretical kt decreases with the increase of h/L, f, and D/h. Good agreement between the experimental and theoretical kt can be observed when the friction factor f is in the range of 5 to 6. The variation of the experimental and theoretical kr with both h/L and f for different values of D/h is shown in Fig. 6. The figure shows that the experimental kr increases with the increase of h/L and D/h. In detail, kr increases from 0.15 to 0.57, 0.17 to 0.67, 0.23 to 0.75 when D/h = 0.2, 0.36, and 0.52, respectively as h/L increases from 0.13 to 0.41. The scatter in the variation of the experimental kr with h/L is larger than the scatter in kt, the coefficient of determination (R2) ranged from 0.90 to 0.93. The scatter in the measured kt and kr are due to mainly to the small deviation

of the measured wave height from the target wave height; this deviation is caused by the multiple reflections between the wall and the wave generator (Huang 2007). In addition, the theoretical kr increases with the increasing values of h/L, f, and D/h. It can be observed from Fig. 6 that a reasonable agreement between the experimental and the theoretical kr exists when the friction factor f ranging from 5 to 6. Fig. 7 shows the variation of the experimental and theoretical energy dissipation of the incident wave (kd), see Eq. (35), with h/L and f for different values of D/h. The theoretical and the experimental results covered h/L ranging from 0.13 to 0.41. Fig. 7 shows that the experimental kd increases from 0.31 to 0.51, 0.36 to 0.53, and 0.42 to 0.55 as h/L increases from 0.13 to 0.3 when D/h = 0.2, 0.36, and 0.52 respectively. Then, it slightly decreases up to 0.48, 0.42 and 0.40 at h/L = 0.41. The scatter in the variation of the experimental kd with h/L is larger than the scatter in kt, the coefficient of determination (R2) ranged from 0.9 to 0.95. The theoretical kd increases with h/L increasing up to certain value (h/L = 0.346, 0.315, and 0.306 for D/h = 0.20, 0.36, and 0.52 respectively) after which, it slightly decreases. Also, the theoretical kd increases with f increasing up to h/L reaches the same value for f = 6 and then changes to the opposite direction. It can be observed from the figure that a reasonable agreement between the experimental and theoretical kd exists when f ranges between 5 and 6. The verification of the theoretical model for estimating the different hydrodynamic coefficients when f = 5.5 is presented in Fig. 8. It can be observed from the figure that a good agreement is obtained between the experimental and theoretical transmission coefficient (kt) and the

Fig. 6. Comparison between the experimental and theoretical results of the reflection coefficient for different f and D/h.

Fig. 7. Comparison between the experimental and theoretical results of the energy dissipation coefficient for different f and D/h.

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

reflection coefficient (kr). In addition, a reasonable agreement is obtained between the experimental and theoretical energy dissipation coefficient (kd) but in some cases, the theoretical model underestimates kd by value not more than 10%. In addition, Fig. 8 presents the fitted linear equation for the theoretical and experimental results of kt, kr, and kd with the regression coefficient R2. Fig. 9 presents the verification of the present theoretical model for estimating the transmission coefficient by the comparison with available theoretical and experimental results obtained by other studies available from literature where the comparison is made using the upper part of the breakwater only (semi-immersed vertical impermeable barrier) when h/L = 0.34, and 0.17, respectively. The figure shows a good agreement for the magnitude of the transmission coefficient obtained by the present theoretical results and those

Fig. 8. Comparison between the experimental and theoretical results of the different hydrodynamic coefficients when f = 5.5.

753

obtained by using Eigenfunction Expansion Method (Abul-Azm, 1993) and by using Boundary Integral Element Method (Liu and Abbaspour, 1982). Also, the present theoretical results agree reasonably well with the experiments of Koraim (2005), Heikal (1997), and Wiegel (1960).

5.3. Hydrodynamic performances comparisons Fig. 10 presents the comparison between the performance of the present vertical wall model and the performance of different models investigated by other authors when Hi/gT2 ranged between 0.0025 and 0.0123. The figure shows considerable scatter in the performance of the different compared models. This can be attributed to the difference in the models geometry and cross sections shape (see Table 1). Also, the transmission coefficient decreases with increasing Hi/gT2, while the reflection and energy dissipation coefficients follow the opposite trend. Fig. 10 shows that the present vertical wall model gives lower values of kt than other authors, except Abdel-Mawla and Balah, 2001 (double rows) . In addition, the present model gives higher values of kr than the others. Fig. 10 also shows that the present vertical wall model dissipates the wave energy in a similar way than described by other authors in most cases. Fig. 11 presents the comparison between the performance of the present vertical wall model and the performance of different models investigated by other authors when h/L ranged between 0.14 and 0.42. The figure shows that the transmission coefficient decreases with increasing h/L, while the reflection coefficient follows the opposite trend. Fig. 11 also shows that the present vertical wall

Fig. 9. Comparison between transmission coefficients obtained by different theories and experiments for the case of the upper part only (semi-immersed barrier).

754

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

Fig. 11. Comparison between the performance of the present model and different similar models of other studies (Hi/gT2 is the main function). Fig. 10. Comparison between the performance of the present model and different similar models of other studies (h/L is the main function).

model gives suitable values of kt, kr, and kd with respect to other models. 5.4. Theoretical results sample Figs. 12 and 13 present a sample of the present theoretical results of the different hydrodynamic coefficients showing the effect of the upper part draft and the lower part porosity (ε), respectively. The theoretical results covered values of h/L ranging from 0.0 to 0.5. Fig. 12 presents the theoretical results for different D/h (D/h = 0.1 to 0.7) when ε = 0.33. The figure shows that kt decreases with the increase of both h/L and D/h while kr follows the opposite trend. Also it can be observed that kd increases to reach the maximum value (kd about 0.5) as h/L increases up to a certain value(h/L = 0.28, 0.28, 0.25, 0.25, 0.2, 0.2, and 0.16 when D/h = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7, respectively) and then start to decrease. In addition, kd increases with the increase of D/h to h/L about 0.2, after which it decreases again. Also, as h/L decreases the effect of the relative upper part draft (D/h) disappears because the long wave is relatively small affected by the vertical wall. Fig. 13 presents the theoretical results for different ε (ε = 0.1 to 0.5) when D/h = 0.5. The figure shows that kt decreases as h/L

increases and ε decreases. This may be due to increasing the screen resistance as ε decreases, and then the percentage of waves, which allowed to be transmitted, decrease. While kr follows the opposite trend as expected. Also, kd increases to about 0.5 as h/L increases up to certain value (h/L = 0.05, 0.15, 0.2, 0.24, and 0.26 when ε = 0.1, 0.2, 0.3, 0.4, and 0.5, respectively), and then starts to decrease. Also, kd increases as ε decreases for h/L b 0.1 and it takes the opposite trend when h/L N 0.2. Also, as the relative wave length (h/L) increases the effect of the lower part porosity (ε) disappears because the wave motion is minimal in the lower part of the water column for short waves (Huang 2007). 5.5. Model application for a full scale case The model results might be illustrated when the model is applied to a real case. In this case, recommended water depth is h = 15 m, incident regular wave height Hi = 2 m, the wave period T = 6 s, and the wave length L = 51 m (h/L = 0.294). Based on Fig. 13 for D/h = 0.5, h/L = 0.294 and, the allowed transmitted wave height Ht = 0.5 m (kt = Ht/Hi = 0.25), then ε = 0.25, and kr = 0.75 then, the recommended breakwater dimensions are; The upper part draft D = 7.5 m (D/h = 0.5); If the slots width w = 0.6 m, the clear distance between each two slots c = 0.20 m (ε = 0.25);

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

755

Fig. 12. Effect of the upper part draft ratio (D/h) on the theoretical results of different hydrodynamic coefficients when ε = 0.33, f = 5.5.

Fig. 13. Effect of the lower part porosity (ε) on the theoretical results of different hydrodynamic coefficients when D/h = 0.5, f = 5.5.

The breakwater width b = 0.6 m (b = w as experimental study); The height of the breakwater above the still water level should be more than 1.70 m (the half of the sea side wave height) in which the maximum resulting wave height at the sea side=Hi *(1+kr)=3.4 m; and This breakwater reduces the incident regular wave from 2 m to 0.5 m (Ht = Hi kt).

the friction factor f = 5.5. Also, the proposed structure type gives high performance when compared with other breakwater systems. The proposed method can be useful for engineers in carrying out preliminary design of vertical wall with horizontal slots subject to normal regular waves by knowing the required degree of protection. In general, the transmission coefficient (kt) decreases with the relative wave length (h/L) and the relative upper part draft (D/h) increasing and with lower part porosity (ε) decreasing while the reflection coefficient (kr) takes the opposite trend. In addition, the wall model dissipates about 50% of the incident wave energy when it is constructed at a water depth equal to about 0.25 to 0.35 of the wave length (h/L = 0.25 to 0.35). In the future, more investigations with different lower part porosities, irregular waves, and obliquely incident waves are required. The extension of the theoretical model to a double vertical breakwater with horizontal slots and the associated laboratory experiments may also be future extensions of the model.

6. Conclusions The wave transmission, reflection, and energy dissipation of a vertical wall with a permeable lower part (horizontal slots) were experimentally and theoretically studied under normal regular waves and no overtopping conditions. The effect of different wave and structural parameters on the hydrodynamic characteristics was investigated e.g. the wave length, the upper part draft ratio, and the lower part porosity. Also, the theoretical model based on an Eigen Function Expansion Method and a Least Square Technique was developed to study the hydrodynamic breakwater performance. In order to examine the validity of the theoretical model, the theoretical results were compared with experimental results and with the results obtained from different studies available from literature (for special case). Comparison between experiments and predictions showed that the theoretical model provides a good estimate of the wave transmission, reflection, and energy dissipation coefficients when

Notations The following symbols have been adopted for use in this paper: A0 An

complex reflection coefficient [−]; complex unknown coefficients [−];

756

b Cm c D Ei Er Et f G g Hi Hr Ht h k kd kr kt L T t w x, z ε ϕp ϕ1, ϕ2 ω

O.S. Rageh, A.S. Koraim / Coastal Engineering 57 (2010) 745–756

breakwater width [m]; added mass coefficient [−]; clear distance between slots [m]; breakwater draft [m]; energy of incident waves [t m]; energy of reflected waves [t m]; energy of transmitted waves [t m]; friction factor [−]; permeability parameter [−]; acceleration of gravity [m/s2]; incident wave height [m]; reflected wave height [m]; transmitted wave height [m]; water depth [m]; incident wave number [1/m]; energy dissipation coefficient [−]; reflection coefficient [−]; transmission coefficient [−]; wave length [m]; wave period [s]; time [s]; width of slot [m]; two dimensional axis [m]; porosity of the lower part of breakwater [−]; total flow velocity potential [−]; seaward and shoreward velocity potential respectively [−]; and angular wave frequency [1/s]

References Abdel-Mawla, S., Balah, M., 2001. Wave energy absorption by an inclined slotted-wall breakwater. Journal of Scientific Research, Faculty of Engineering, Suez Canal University, Port Said. Abul-Azm, A.G., 1993. Wave diffraction through submerged breakwaters. Journal of Waterway, Port, Coastal and Ocean Engineering 119 (6), 587–605. Balaji, R., Sundar, V., 2002. Hydraulic performance of vertical wave screens with pipes. Proceeding of 10th International Maritime Association of the Mediterranean, Crete, Hellas, Tech. Session 9, paper No: 127. Balaji, R., Sundar, V., 2004. Theoretical and experimental investigation on the wave transmission through slotted screens. Oceanic Engineering International 8 (2), 69–90. Bergmann, H. 2000. Hydraulische wirksamkeit und seegangsbelastung senkrechter wellenschutzbauwerke mit durchlässiger front. Ph.D. Thesis, Mitteilungen aus dem Leichtweiss-Institut, TU Braunschweig (in German). Bergmann, H., Oumeraci, H., 1998. Wave pressure distribution on permeable vertical walls. Proceedings of the 26th Int. Conf. on Coastal Eng. Copenhagen, Denmark. Clauss, G.F., Habel, R., 1999. Hydrodynamic characteristics of underwater filter system for coastal protection. Proceeding of the Canadian Coastal Conference, Victoria, Canada, vol. 1, pp. 139–154.

Dalrymple, R.A., Martin, P.A., 1990. Wave diffraction through offshore breakwaters. Journal of Waterway, Port, Coastal and Ocean Engineering 116 (6), 727–741. Galal, S., 2002. The use of permeable breakwater for sea defense and shore protection. MSc. Thesis, Faculty of Engineering, Suez Canal University, Port-Said. Goda, Y., Suzuki, Y., 1976. Estimation of incident and reflected waves in random wave experiments. Proceeding of the 15th Costal Engineering Conference, pp. 828–845. Heikal, E.M., 1997. Wave transmission over sloping beaches behind floating breakwaters. Journal of Scientific Bulletin, 32. Ain-Shams University, pp. 195–208. Huang, Z., 2007. Wave interaction with one or two rows of closely spaced rectangular cylinders. Ocean Engineering 34, 1584–1591. Isaacson, M., Premasiro, S., Yang, G., 1998. Wave interaction with vertical slotted barrier. Journal of Waterway, Port, Coastal and Ocean Engineering 124 (3), 118–126. Isaacson, M., Baldwim, J., Premasiro, S., Yang, G., 1999. Wave interaction with double slotted barriers. Applied Ocean Research 21, 81–91. Koether, G. 2002. Hydraulische Wirksamkeit und Wellenbelastung getauchter Eiinzelfilter und Unterwasserfiltersysteme für den Küstenschutz, Ph.D. Thesis, TU Braunschweig, Leichtweiss-Institut für Wasserbau, (in German) Koether, G., Bergmann, H., Oumeraci, H., 2000. Wave attenuation by submerged filter systems. Proceeding. of 4th. International Conf. on Hydro-dynamics (ICHD), vol. II, pp. 711–716. Yokohama, Japan. Koraim, A. S., 2005. Suggested model for the protection of shores and marina. Ph.D. Thesis in Civil Engineering, Zagazig University, Zagazig, Egypt. Koutandos, E.V., 2009. Hydrodynamics of vertical semi-immersed slotted barrier. WSEAS Transaction on Fluid Mechanics 4 (3), 85–96. Kriebel, D.L., 1992. Vertical wave barriers: wave transmission and wave forces. Proceeding of the 23th International Conf. on Coastal Eng., New York, pp. 1313–1326. Kriebel, D.L., 2004. Design methods for timber wave screens. Report EW-01-04, U.S Naval Academy. Ocean Eng. Program, pp. 3891–3903. Krishnakumar, C., Balaji, R., Sannasiraj, S.A., Sundar, V., 2008. Reflection and transmission characteristics of partially submerged slotted wave screens. International Journal of Ecology and Development, Special Issue on Coastal Environment: Coastal processes 11 (F08), 20–35. Laju, k., Sundar, v., Sundaravadivelu, R., 2007. Studies on pile supported double skirt breakwater models. Journal of Ocean Technology 2 (1), 32–53. Liu, P.L.F., Abbaspour, M., 1982. Wave scattering by a rigid thin barrier. Journal of Waterway, Port, Coastal and Ocean Engineering 108 (4), 479–491. Mani, J.S., 1998. Wave forces on partially submerged pipe breakwater. Ocean Wave Kinematics Dynamics and Loads on Structures 1, 505–512. Mani, J.S., Jayakumar, S., 1995. Wave transmission by suspended pipe breakwater. Journal of Waterway, Port, Coastal and Ocean Engineering 121 (6), 335–338. Rao, S., Rao, N.B.S., Shirlal, K.G., Reddy, G.R., 2003. Energy dissipation at single row of suspended perforated pipe breakwaters. Journal of Institution of Engineers 84, 77–81 CV. Reddy, M.S., Neelamanit, S., 1992. Wave transmission and reflection characteristics of a partially immersed rigid vertical barrier. Ocean Engineering 19, 313–325. Sollitt, C.K., Cross, R.H., 1972. Wave transmission through permeable breakwaters. Proceeding: 13th Coastal Engineering Conference. ASCE, Vancouver, pp. 1827–1846. Suh, K.D., Shin, S., Cox, D.T., 2006. Hydrodynamic characteristics of pile-supported vertical wall breakwaters. Journal of Waterway Port, Coastal and Ocean Engineering 132 (2), 83–96. Suh, K.D., Jung, H.Y., Pyun, C.K., 2007. Wave reflection and transmission by curtain wall– pile breakwaters using circular piles. Ocean Engineering 34, 2100–2106. Sundar, V., Subbarao, B.V.V., 2002. Hydrodynamic pressures and forces on quadrant front face pile supported breakwater. Ocean Engineering 29, 193–214. Ursell, F., 1947. The effect of a fixed vertical barrier on surface waves in deep water. Proceeding of the Cambridge Philosophical Society 43 (3), 374–382. Wiegel, R.L., 1960. Transmission of wave past a rigid vertical thin barrier. Journal of Waterways and Harbor Division 86 (1), 1–12. Yu, X.P., 1995. Diffraction of water waves by porous breakwaters. Journal of Waterway Port, Coastal and Ocean Engineering 121 (6), 275–282.