the use of vertical walls with horizontal slots as breakwaters · the use of vertical walls with...

Thirteenth International Water Technology Conference, IWTC 13 2009, Hurghada, Egypt

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THE USE OF VERTICAL WALLS WITH HORIZONTAL SLOTS AS

BREAKWATERS

O. S. Rageh 1 and A. S. Koraim 2

1 Associate Prof., Irrigation and Hydraulic Dept., Faculty of Engineering, Mansoura University, El-Mansoura, Egypt

2 Dr., Water and Water Structure Engineering Dept., Faculty of Engineering, Zagazig University, Zagazig, Egypt

ABSTRACT The hydrodynamic efficiency of the breakwater which consists of one row of vertical wall suspended on supporting piles was experimentally and theoretically studied. The wall was divided into two parts; the upper part extending above the water level to some distance below sea level was impermeable. The other part of the wall was permeable which is consisted of closely spaced horizontal slots. The theoretical model was based on the linear wave theory, an Eigen function expansion method and the least square technique. The efficiency of the breakwater was presented experimentally and theoretically as a function of the transmission, the reflection and the wave energy loss coefficients for different wave and structure parameters. To examine the validity of the theoretical model, the theoretical results were compared with the present experimental results and results obtained from other studies. The results indicated that the proposed theoretical model can be accepted for predicting the different hydrodynamic coefficients when the friction factor ranged from 5 to 6. 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 is arisen to protect and maintain these coasts against the destructive forces of nature such as waves and currents. It is necessary to consider cost-effective and environment friendly structures that decrease the effect of these forces 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 conditions. 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.

Thirteenth International Water Technology Conference, IWTC 13 2009, Hurghada, Egypt ��

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 extending above the water level to some distance below sea level. The other part of the wall is permeable which is consisted of closely spaced horizontal slots. This type helps in dissipating energy of the sea incident wave and protecting shorelines from erosion and wave attack. It also, possesses desirable features such that minimizing the pollution aspects near shores because it permits the flow exchange between the partially enclosed water body and the open sea and experience slightly less total hydrodynamics force compared with vertical solid breakwater. There is very little material available on this type of breakwater. Many experimental and theoretical studies were carried out determining the efficiency of models similar to the upper part and the lower part of the proposed model individually. The efficiency of the upper part (semi-immersed impermeable wall) used as a breakwater was experimentally and theoretically studied by many researchers. Ursell (1947), Weigel (1960), Reddy and Neelamani (1992), Heikal (1997), and Koraim (2005) carried out experimental studies to determine the efficiency of this type. Liu and Abbaspour (1982) used a Boundary Integral Element Method. Abul-Azm (1993) and Heikal (1997, 2007) developed theoretical models using the Eigen Function Expansion Method to determine the efficiency of this type of breakwaters. The functional performance of the slotted or screen breakwater was evaluated by examining the wave reflection and transmission through the breakwater. In order to examine the wave scattering by vertical slotted breakwaters, physical hydraulic tests and many analytical models were developed such as Kriebel (1992), Isaacson et al. (1998), Clauss and Habel (1999), Isaacson et al. (1999), Abdel-Mawla and Balah (2001), Balaji and Sundar (2002), Huang (2007), and Koraim (2007). The hydrodynamic performance of the pipe breakwater was evaluated by Mani and Jayakumar (1995), Mani (1998), Galal (2002), and Rao et al. (2003). The interaction between the impermeable upper part and the slotted lower part was investigated recently by Sundar and Subbarao (2002). Sundar and Subbarao (2003) carried out the experimental studies on the quadrant front face pile supported breakwaters. The bottom portion consisted of closely spaced piles and the top portion consisted of a quadrant solid front face on the seaside. The leeward side of the top portion was with a vertical face. They measured the different hydrodynamic coefficients (transmission, reflection, and wave energy loss coefficients). In addition, they measured the hydrodynamic pressure and forces exerted on this breakwater model. Suh et al. (2006, 2007) studied experimentally and theoretically the hydrodynamic characteristics of a curtain-wall-pile breakwater. The upper part of this model was a


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vertical wall and the lower part consisted of an array of vertical square and circular piles. Mathematical models were developed using the Eigen function expansion method to compute wave transmission, reflection, run up, and wave force acting on the breakwater for both regular and irregular waves. Large-scale laboratory experiments also carried out for curtain-wall-pile breakwaters with a constant spacing between piles but various drafts of the upper vertical wall. Laju et al. (2007) studied the pile supported skirt breakwater. The breakwater model consisted of an impermeable wave barrier near the free surface supported on steel or concrete piles. The impermeable vertical barrier extended above the water level to some distance below sea level. They described the details of the numerical and experimental studies on the hydrodynamic characteristics of this type of breakwater. The numerical model was based on Eigen function expansion theory for linear waves. In this paper, the hydrodynamic efficiency of the vertical wall of permeable lower part (horizontal slots) which was suspended on supporting piles was experimentally and theoretically studied. The breakwater efficiency was measured and computed as function of transmission, reflection, and wave energy loss coefficients. The effect of different wave and structural parameters on the breakwater efficiency such as; the wave length, the wave period, the upper part draft ratio, and the lower part porosity were investigated. Also, the theoretical model based on Eigen function expansion method and the least square technique was used to study the hydrodynamic breakwater efficiency. 2. THEORETICAL MODEL Let us consider the breakwater sketched in Fig. (1), in which h is the constant water depth in still water; D is the draft of the upper part; b is the breakwater thickness; w is the slot width; c 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 is incident in the positive x -direction. The fluid domain is divided into two regions. Region 1 located seaward the breakwater at x � 0 and region 2 located shoreward the breakwater at x � 0. 2.1. 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)] may be


expressed for the sea side and the shore side regions as follows, Issacson et al. (1998) and Suh et al. (2006):

]e)z,x()khcosh(

12

igHRe[)t,z,x( ti

pi

pω−φ

ω−=φ p = 1, 2 (1)

in which Re is the real part of the complex expression between the brackets, 1i −= , g is the acceleration of gravity, ω is the angular wave frequency (ω=2π/T), T is the wave period, k is the wave number (k=2π/L) , L is the wave length, and p = 1, 2 refers to the two wave regions at seaward and shoreward, respectively.

Figure (1) Schematic diagram for the breakwater model Assuming that the wall thickness is very small compared with the wave length, so that the wall has no thickness mathematically. Then φP(x, z) must satisfy the following matching conditions at x = 0:

xx21

∂φ∂=

∂φ∂

= 0 for 0 > z > -D (2)


��

)(iGxx 12

\21 φ−φ−=∂φ∂=

∂φ∂

for -D > z > -h (3)

The first matching condition, Equ. (2) describes that the horizontal velocities which vanish on both sides of the upper impermeable part of the breakwater. The second one, Equ. (3), for the lower part of the breakwater describes that the horizontal velocities in the two regions must be the 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 G\=G/b, 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 constant G. In the present study, the method of Sollitt and Cross (1972), and Isaacson et al. (1998) is adopted and G is expressed by:

isG

−ε=

f (4)

in which, ε is the porosity of the lower part [ε =c/(c+w)], f is the friction coefficient which is calculated implicitly using the Lorentz principle of equivalent work, and s is the inertia coefficient which is given by:

)1

(c1s m εε−+= (5)

in which, Cm is the added mass coefficient. Cm is treated as a constant (Cm=0) as suggested by Issacson et al. (1998). 2.2. Flow Potential Solution The reduced velocity potentials seaward, φ1, and shoreward, φ2, are obtained using the Eigen function expansion method as given by Issacson et al. (1998) and Suh et al. (2006). The velocity potentials are expressed in a series of infinite number of solutions as follows:

xn

0nnI1

ne)hz(cosA)z,x( µ∞

=+µ−φ=φ � (6)

xn

0nnI2

ne)hz(cosA)z,x( µ−∞

=+µ+φ=φ � (7)

where, φI is the incident wave potential which is given as:


ikx

I e)hz(kcosh)z,x( +=φ (8) Also, An are unknown complex coefficients and µn for n � 1 for non-propagating evanescent waves are the positive real roots of the following dispersion relation, taken in ascending order:

)htan(g nn2 µµ−=ω for n � 1 (9)

�0 itself corresponds to the imaginary root of the above equation for propagating waves, such that �0=-ik, with the wave number k being given as the real root of the corresponding equation:

)khtanh(gk2 −=ω for n = 0 (10) 2.3. Dual Series Relations: Substituting Equ. (6) and Equ. (7) into the boundary conditions at the breakwater, Equ. (2) and Equ. (3), when x=0, yields:

)hz(cos)hz(cosA0n

00nnn +µµ−=+µµ�∞

= 0>z>-D (11)

)hz(cos)hz(cos)iG2(A 000n

n\

nn +µµ−=+µ−µ�∞

= -D>z>-h (12)

Equations (11) and (12) are known as dual series relations as given by 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 specifies the potential along z-axis as follows:

)hz(cos)hz(cosA)z(G0n

00nnn +µµ++µµ= �∞

= 0>z>-D (13)

)hz(cos)hz(cos)iG2(A)z(G 000n

n\

nn +µµ++µ−µ= �∞

= -D>z>-h (14)

2.4. Solution of the System of Equations: The least square technique, Dalrymple and Martin (1990), may be used to determine the coefficients An which requires the value of:


��

�−

=0

h

2 0dz)z(G to be minimum (15)

Minimizing this integral with respect to each of the An coefficients leads to the following equation:

�−

=∂∂0

hm

* 0dzA/)z(G)z(G m=0,1,2,…… (16)

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

)hz(cos)hz(cosA)z(G0n

0*0n

*n

*n

* +µµ++µµ= �∞

= 0>z>-D (17)

)hz(cos)hz(cos)iG2(A)z(G 0*0

0nn

*\*n

*n

* +µµ++µ+µ= �∞

= -D>z>-h (18)

in which, An

*, µn*, µ0

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

)hz(cosA/)z(G mmm +µµ=∂∂ 0>z>-D (19)

)hz(cos)iG2(A/)z(G m\

mm +µ−µ=∂∂ -D>z>-h (20) substituting into Equ. (16) using Equ. (17) to Equ. (20), then:

m00n

nm*nA α=η�

∞

= (21)

in which:

)]D,h(S)iG2)(iG2()0,D(S[ nm\

m*\*

nnmm*nnm −−−µ+µ+−µµ=η (22)

)]D,h(S)iG2()0,D(S[ m0\

mm0m*0m0 −−−µ−−µµ−=α (23)

where:

dz)hz(cos)hz(cos)q,p(Sq

pmnnm � +µ+µ=

q

pmn

mn

mn

mn)(

)zh)(sin()(

)zh)(sin(21

��

��

�

µ−µ+µ−µ+

µ+µ+µ+µ= n, m = 0, 1, 2,.. (24)


Equation (21) can be transformed in the matrix form as follows:

��

�

�

��

�

�

α

αα

=

��

�

�

��

�

�

��

�

�

��

�

�

ηηηη

ηηηη

ηηηη

m0

01

00

*n

*1

*0

nm2n1n0n

m1121110

m0020100

.

.

A

.

.

A

A

...........

.........

.........

(25)

By solving Equation (25) for evanescent wave modes, N, the equation turn to a set of N*N linear simultaneous equations in unknown complex coefficients A*

n. These equations are solved using a FORTRAN program based on standard matrix inversion technique (Gauss-Elimination Technique). Then; the potential functions representing the different regions can be determined and the reflection and the transmission coefficients can also be determined. 2.5. Reflection, Transmission and Energy Loss Coefficients The theoretical reflection and transmission coefficients (kr and kt) related to the first term of An coefficients (A0) and can be evaluated as follows:

kr = |A0| (26) kt = |1 + A0| (27)

To get a simple formula for A0, the propagating wave mode only (n=m=0) is considered. By substituting n=m=0 in Equation (21), A*

0 which is the complex conjugate of A0 can be obtained as follows:

00

00*0A

ηα

= (28)

where;

)D,h(S)G2k)(G2k()0,D(Sk 00\*\

002

00 −−+++−=η (29)

)]D,h(S)G2k(k)0,D(Sk[ 00\

002

00 −−++−−=α (30)

�−

+=−0

D

200 dz)hz(kcosh)0,D(S

��

��

� +−−= Dk2

)Dh(k2sinhkh2sinh21

(31)


��

�−

−+=−−

D

h

200 dz)hz(kcosh)D,h(S

��

�

��

�−+

−= Dh

k2

)Dh(k2sinh

21

(32)

Theoretically, energy equilibrium of an incident wave energy attack the structure can be expressed as follows:

Ei = Er + Et (33) in which, Ei is the energy of incident wave, Er is the energy of reflected wave, and Et is the energy of transmitted wave. Also, Equ. (33) can be rewritten as follows:

1kk 2t

2r =+ (34)

Actually, when the wave reaches the structure, some of the wave energy is dissipated by the structure itself. This part of the wave energy can be estimated as a function of the reflection and transmission coefficients as given by Issacson et al. (1998):

2t

2rL kk1k −−= (35)

in which, kL is the wave energy loss coefficient. 3. EXPERIMENTAL WORK A series of experiments were conducted in the Irrigation and Hydraulic Laboratory, faculty of engineering, El-Mansoura University. The tests were carried out to measure the hydrodynamic coefficients of wave scattering by the suggested breakwater model using different wave and structural parameters. The dimensions of the wave flume were 15m long, 1m width, and 1m 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 50cm and with generator motions corresponding to regular wave trains with different wave periods ranged from T=0.9 to 1.9sec. The suggested breakwater model was consisted of two parts. The upper part was impermeable with constant thickness (b=2.5cm) and with variable draft (D=10.0, 18.0, and 26.0cm). The lower part was permeable and consisted of closely spaced horizontal slots of fixed porosity (ε=0.5). The slot width (w=2.0cm), slot thickness (b=2.5cm), and the clear distances between slots (c=2.0cm) were constant. The breakwater model were placed at the middle of the wave flume.


The wave period, length and height were recorded by using Sony MVC-CD 500 Digital Still Camera. To measure the incident (Hi) and reflected (Hr) wave heights, two recording positions (P2 and P1) were determined in front of the breakwater model (wave generator side) at distances 0.2L and 0.7L respectively. This is according to the two-point method of Goda and Suzuki (1976). To measure the transmitted (Ht) wave heights, one recording position (P3) was determined behind the breakwater model (wave absorber side) at a distance 1.5m.

Figure (2) Details of wave flume, model position and wave recording locations

4. RESULT ANALYSIS AND VERIFICATION: About 45 experimental tests were carried out to check the validity of the above-mentioned theoretical model in predicting the transmission, reflection and wave energy loss coefficients. The results of these tests were presented as figures in form of dimensionless parameters. The results of the theoretical model for the same parameters also were presented on these figures. Figure (3) presents the variation of the experimental and theoretical transmission coefficient (kt) with the relative wave length (h/L) and friction factor (f) for different breakwater draft ratios (D/h). The figure shows that, the experimental kt decreases with the increasing values of both h/L and D/h. In which, 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 in which the R2=coefficient of determination ranged from 0.97 to 0.99. The figure shows that, the theoretical kt decreases with the increase of both h/L, f and D/h. Good agreement between the experimental and theoretical kt can be observed when the friction factor f ranging from 5 to 6.


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Figure (3) Comparison between the experimental and theoretical results of the transmission coefficient for different values of f and D/h


The variation of the experimental and theoretical kr with both h/L and f for different values of D/h is shown in Figure (4). The figure shows that, the experimental kr increases with the increase of h/L and D/h. In which, 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 more than the scatter in kt in which the coefficient of determination (R2) ranging from 0.90 to 0.93. The figure shows that, the theoretical kr increases with the increasing values of h/L, f, and D/h. It can be observed from the figure that a reasonably agreement between the experimental and theoretical kr when the friction factor f ranging from 5 to 6. Figure (5) shows the variation of the experimental and theoretical energy loss of the incident wave (kL) 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. The figure shows that, the experimental kL 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 as h/L=0.41. The scatter in the variation of the experimental kL with h/L is more than the scatter in kt. In which the coefficient of determination (R2) ranging from 0.9 to 0.95. The theoretical kL increases with h/L increasing up to certain value then, it slightly decreases. Also, the theoretical kL increases with f increasing up to h/L reaches certain value then, it takes the opposite direction. It can be observed from the figure that a reasonable agreement between the experimental and theoretical kL when f ranging from 5 to 6. The comparison between the present experimental and the theoretical results of the different hydrodynamic coefficients when f=5 and 6 is presented in Figure (6). It can be observed from the figure that, a good agreement is obtained between the experimental and theoretical transmission coefficient (kt) when the friction factor f =6. While a good agreement between the experimental and theoretical reflection coefficient (kr) is obtained when f =5. In addition, a reasonable agreement is obtained between the experimental and theoretical energy loss coefficient (kL) when f =5 or 6. But in some cases, the theoretical model underestimates kL by value not more than 10%. Figures (7) and (8) present the comparison among the present theoretical transmission coefficient and other theoretical and experimental results obtained by other studies. This is for using the upper part only of the breakwater for different draft ratios (D/h) when h/L=0.34, and 0.17 respectively. The figures show a good agreement for the magnitude of the transmission coefficient obtained by the present theoretical results and those obtained by using Eigen Function 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).


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Figure (4) Comparison between the experimental and theoretical results of the reflection coefficient for different values of f and D/h


Figure (5) Comparison between the experimental and theoretical results of the energy loss coefficient for different values of f and D/h


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Figure (6) Comparison between the experimental and theoretical results of the different hydrodynamic coefficients


Figure (7) Comparison between transmission coefficients obtained by different theories and experiments for upper part only when h/L=0.34

Figure (8) Comparison of transmission coefficients obtained by different theories and experiments for upper part only when h/L=0.17


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Figures (9) and (10) 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 ranged from 0.0 to 0.5. The number of evanescent wave modes (n and m) in the theoretical model kept at 20. Figure (9) 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, kL increases to reach the maximum value (kL about 0.5) as h/L increases up to certain value, then start to decrease. This value lies between h/L=0.15 and 0.25 according to D/h. In addition, kL increases with the increase of D/h to h/L about 0.2, then it takes the opposite trend. Figure (10) 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 while kr follows the opposite trend. Also, kL 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), then start to decrease. Also, kL increases as ε decreases for h/L<0.1 and it take the opposite trend when h/L>0.2. 5. CONCLUSIONS The main conclusions of the present study can be summarized as follows: 1. The proposed theoretical model provides a good estimate the transmission

coefficient when f = 6. 2. The proposed theoretical model gives a good estimate the reflection coefficient

when f = 5. 3. The proposed theoretical model reasonably estimate the wave energy loss

coefficient when f = 5 or 6. 4. The proposed theoretical model gives a good agreement with other theoretical

and experimental studies. 5. The tested breakwater system dissipates about 50% from the incident wave

energy when it is constructed at water depth equal about 0.25 to 0.35 of the wave length (h/L=0.25 to 0.35).


Figure (9) Effect of upper part draft (D) on the theoretical results of different hydrodynamic coefficients when εεεε=0.33, f=6 and n=20


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Figure (10) Effect of lower part porosity (εεεε) on the theoretical results of different hydrodynamic coefficients when D/h = 0.5, f=6 and n=20


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NOTATION

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

A0 = complex reflection coefficient; An = complex unknown coefficients; b = breakwater width; Cm = added mass coefficient; c = clear distance between slots; D = breakwater draft; d = pipes diameter; Ei = energy of incident waves; Er = energy of reflected waves; Et = energy of transmitted waves; f = friction coefficient;


G = permeability parameter; g = acceleration of gravity; Hi = incident wave height; Hr = reflected wave height; Ht = transmitted wave height; h = water depth; i = imaginary number ( − 1 ); k = incident wave number; kL = energy loss coefficient; kr = reflection coefficient; kt = transmission coefficient ; L = wave length; m = number; n = number; R2 = correlation factor; T = wave period; t = time; w = width of slot; x, z = two-dimensional axis; ε = porosity of the lower part of breakwater; φp = total flow velocity potential; φ1, φ2 = seaward and shoreward velocity potential respectively; and ω = angular wave frequency.

the use of vertical walls with horizontal slots as breakwaters · the use of vertical walls with...

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