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Abstract

The purpose of this study is to estimate the wave height located at the front face of the offshore refracted breakwater when the trench is dredged in the exterior region of the breakwater at various depths. The problems that arise with regard to the different obliquely incident wave field involve having two-dimensional planes and having the configuration of the trench region designated by a single horizontal rectangular system. The numerical approach involves applying the Green function based on the boundary integral approach. The results of the present numerical works are illustrated. The ratio of the wave height reduction at the front face of the refracted breakwater is shown to be more than 25% due to the trench’s effect on the seabed. The corner of the refracted breakwater is where the wave energy is concentrated, and through utilizing the trench, the wave energy can be subdued to reduce the damage on the breakwater. The trench brings about a reduction in wave amplitude and wave energy in, around, and directly downstream of the trench.

Keywords

Green function wave energy wave amplitude dredge

Introduction

The incident water waves that propagate from the offshore zone concentrate at the corner of the breakwater. Water waves of powerful energy, namely, storm surges and high waves, can damage the breakwater, and therefore, affect the safety of the breakwater. Many researchers are investigating new strategies to reduce these powerful wave energies. Various methods are being applied specifically for the reduction of such energy at the corner of the breakwater. One example would be to construct the section of a corner into a slit type, absorbed type, or a curved type.^1–4

The purpose of this study is to reduce the incident wave energy before it reaches the corner of the breakwater by utilizing the dredged region (trench region). When the high-energy sourced waves pass over the trench with discontinuity water depth, the high wave energy level can be reduced.^5,6 This study involves investigating the interaction between the trench and the structure and simulating the reduction effect of the wave energy through the trench.

The early pioneers to research on the interaction of incident waves with a dredged region were notably Newman,⁷ Hilay,⁸ Lee and Ayer,⁹ Miles,¹⁰ Kirby and Dalrymple,¹¹ Ting and Raichlen,¹² and Kirby et al.¹³ The x–z vertical two-dimensional (2D) approach was applied to perform all these studies. To deal with cases where a single or multiple trenches are present, Williams,¹⁴ McDougal et al.,⁵ and Williams and Vazquez⁶ presented a numerical model that involved the x–y horizontal planes. Based on Williams’ method,¹⁴ Takezawa et al.¹⁵ used various configurations to investigate diffractions for the trench breakwater and present wave propagation while the trench was dredged at the port’s entrance.

Briggs and Sargent¹⁶ conducted experiments at the field work at the entrance channel. Later, Briggs et al.¹⁷ also assessed the deep draft channel design. To deal with random waves, Kirby¹⁸ proposed several techniques in Barbers Point Harbor, Hawaii. Reniers et al.¹⁹ developed a computer program for the analysis of long waves at the trench region. Recently, Kim²⁰ and Kim and Lee²¹ investigated the modeling wave height of random waves produced by the submarine trench.

This study is, first of all, about the interaction of waves that are propagated over the dredged trench geometry with discontinuity water depth, and second, about the effect of this wave interaction that takes place on the front face and the vicinity of the offshore refracted breakwater systems when the trench is dredged.

There are three boundary interactions presented concerning this wave interaction: the interaction of discontinuity water depth boundary and the trench boundary, the discontinuity water depth boundary within the trench and the refracted breakwater boundary, and the trench boundary and the breakwater boundary. The problem is considered in a 2D plane; the configuration of the dredging region on the seabed with different depths is a single long rectangular type and the configuration of the breakwater is refracted with the same length between both sides of the breakwater. The numerical simulation utilizes the linear wave theory with the boundary integral equation based on the Green function, and the conditions between the exterior and the interior trench regions and the exterior region of refracted breakwater are matched.

To verify the present numerical model, the results of the experimental works of Koji and Mutsuo²² were compared for the regular wave height at the front face of the refracted breakwater without the trench. The comparing results between the two revealed that the present numerical simulation and the research by Koji and Mutsuo²² fall into good agreement.

During this study, the scattered wave field in the vicinity of the breakwater due to the trench with discontinuity water depth exhibited decreasing effects. This study can offer vital information to help design a dredge line of the outer breakwater to be more distantly located from the coastal zone. Based on the results, the present numerical model proves to be effective and can be adequately utilized to analyze wave interactions by the trench, and thus be an applicable tool in coastal engineering.

Theoretical development

The geometry of the problem is presented in Figure 1. The fluid domain is divided into two regions: an interior trench region $(Ω_{1})$ with varying depths $(d_{2}, d_{3}, d_{4})$ and the exterior fluid region $(Ω_{2})$ with uniform depth $d_{1}$ . The boundary regions express $Γ_{1}$ (trench boundary) and $Γ_{2}$ (refracted breakwater region). The points shown in front of the refracted breakwater in Figure 1 are places to achieve numerical calculation. The Cartesian coordinates shown in Figure 1 are assigned an origin at the corner of the refracted breakwater. The x and y axes are directed in the horizontal plane, and the z-axis is directed vertically upward in relation to the equilibrium water surface.

Figure 1.

Definition sketch for fluid domain and boundaries.

Assuming that the fluid region is taken to be inviscid, incompressible, and the flow irrotational, then the fluid motion may be described in terms of velocity potential $Θ_{j} (x, y, z; t) = Re [Φ_{j} (x, y . z) e^{- i ω t}]$ , where Re [ ] denotes the real part of a complex expression, for j = 1, 2. The governing equation in each of the fluid regions may be written as

\nabla^{2} Φ_{j} = 0 in Ω_{j}; j = 1, 2

(1)

It is subjected to the usual boundary conditions on the free surface and seabed

\frac{\partial Φ_{j}}{\partial z} - \frac{ω^{2} Φ_{j}}{g} = 0 on z = 0

(2)

\frac{\partial Φ_{j}}{\partial z} = 0 on z = - d_{j}

(3)

for j = 1, 2, where $ω$ is the angular frequency and $g$ is the gravitational acceleration.

The Helmholtz equation must satisfy the equations governed in each fluid region

\frac{\partial^{2} Φ}{\partial x^{2}} + \frac{\partial^{2} Φ}{\partial y^{2}} + k_{j}^{2} Φ = 0; j = 1, 2

(4)

where the wave numbers $k_{j}$ (j = 1, 2) are defined by

k_{j} = \frac{ω^{2}}{g \tanh (k_{j} d_{i})}; j = 1, 2

(5)

Continuity of pressure and velocity across the fluid interface between the interior region and the exterior region requires the following conditions to be satisfied

d_{1} \frac{\partial Φ_{1}}{\partial n} = d_{i} \frac{\partial Φ_{2}}{\partial n}; i = 2, 3, 4 on Γ_{1}

(6)

Φ_{1} = Φ_{2} on Γ_{1}

(7)

For the case without the trench, along the vicinity of refracted breakwater $(Γ_{2})$ , continuities of pressure and velocity require that

\frac{\partial Φ_{1}}{\partial n} = \frac{\partial Φ_{3}}{\partial n} on Γ_{2}

(8)

Φ_{1} = Φ_{3} on Γ_{2}

(9)

where $Φ_{3} = Φ_{I} + Φ_{s}$ , where $Φ_{s}$ denotes the scattered velocity potentials and $Φ_{I}$ is the incident potential which is given by

Φ_{I} (x, y, z) = - \frac{{igH}_{I}}{2 ω} \frac{\cosh [k_{3} (z + d_{1})]}{\cosh (k_{3} d_{1})}

(10)

Finally, at large radial distances $r$ , the fluid potential in $Ω_{3}$ has a scattered component subjected to a radiation condition, which may be written as

lim_{r \to \infty} \sqrt{r} (\frac{\partial}{\partial r} - {ik}_{j}) (Φ_{3} - Φ_{I}) = 0; j = 1, 2

(11)

Green’s function $G_{j} {(x', y'), (x, y)}$ , j = 1, 2, may now be expressed for each region as

G_{j} {(x', y'), (x, y)} = \frac{i π}{2} H_{0}^{(1)} (k_{j} R); j = 1, 2

(12)

where $H_{0}^{(1)}$ is the Hankel function of the first kind of order zero and $R = \sqrt{{(x' - x)}^{2} + {(y' - y)}^{2}}$ . $(x, y)$ represents the arbitrary points in the fluid regions, whereas $(x', y')$ shows the points of boundaries in the refracted breakwater and the trench.

Applying Green’s second identity to $Φ_{j}$ and $G_{j}$ , and extending to full regions, we have the following two boundary integral equations

\begin{matrix} Φ_{2} (x', y') & = - \frac{1}{π} \int_{S_{j}} [Φ_{2} (x, y) \cdot \frac{\partial G_{j}}{\partial n} - G_{j} \cdot \frac{\partial Φ_{2}}{\partial n} (x', y')] {dS}_{j}; \\ j = 1, 2 \end{matrix}

(13)

\begin{matrix} Φ_{2} (x', y') = 2 Φ_{I} (x', y') \\ + \frac{1}{π} \int_{S_{j}} [Φ_{2} (x', y') \cdot \frac{\partial G_{j}}{\partial n} - G_{j} \cdot \frac{\partial Ψ_{2}}{\partial n} (x', y')] {dS}_{j}; j = 1, 2 \end{matrix}

(14)

When we apply the boundary conditions (6)–(9) proposed by Williams¹⁴ and McDougal et al.¹⁵ to equations (13) and (14), the following occurs to both equations (15) and (16)

\begin{matrix} Φ_{2} (x', y') = \\ - \frac{1}{π} \int_{S_{1}} [Φ_{2} (x, y) \cdot \frac{\partial G_{1}}{\partial n} - \frac{d_{1}}{d_{i}} G_{1} \cdot \frac{\partial Φ_{2}}{\partial n} (x', y')] {dS}_{1} \\ - \frac{1}{π} \int_{S_{2}} [Φ_{2} (x, y) \cdot \frac{\partial G_{2}}{\partial n} - \frac{d_{1}}{d_{i}} G_{2} \cdot \frac{\partial Φ_{2}}{\partial n} (x', y')] {dS}_{2}; \\ i = 2, 3, 4 \end{matrix}

(15)

\begin{matrix} Φ_{2} (x', y') = 2 Φ_{I} (x', y') \\ + \frac{1}{π} \int_{S_{1}} [Φ_{2} (x, y) \cdot \frac{\partial G_{1}}{\partial n} - \frac{d_{1}}{d_{i}} G_{1} \cdot \frac{\partial Φ_{2}}{\partial n} (x', y')] {dS}_{1} \\ + \frac{1}{π} \int_{S_{2}} [Φ_{2} (x, y) \cdot \frac{\partial G_{2}}{\partial n} - \frac{d_{1}}{d_{i}} G_{2} \cdot \frac{\partial Φ_{2}}{\partial n} (x', y')] {dS}_{2}; \\ i = 2, 3, 4 \end{matrix}

(16)

The free surface elevation $η (x, y, t) = Re {Δ_{j} (x, y) e^{- i ω t}}$ in each region can be obtained from $Φ_{j}$ using the following free surface boundary condition

Δ_{j} (x, y; t) = \frac{iw Φ_{j}}{g}; j = 1, 2

(17)

Finally, the diffraction coefficient of regular wave, $K_{d}$ , is defined as the ratio of the wave height at any point within the interior and the exterior regions, which is given as

K_{d} = | \frac{Φ_{j} (x', y')}{Φ_{I} (x', y')} |

(18)

Validation of the numerical model

Verification of the present numerical model is performed by simulating regular wave diffraction at the vicinity of the breakwater with the trench on the seabed or without the trench. The present numerical results of the wave height at the front face of refracted breakwater were compared with experimental works carried out by Koji and Mutsuo.²²

The conditions of the experimental work and the configuration of the refracted breakwater in regular waves without the trench were set by Koji and Mutsuo²² and are stated as such; the incident wave angle $θ = 90 \circ$ , angle of refracted breakwater $β = 120 \circ$ , reflection coefficient $K_{r} = 0.8$ , and the length of breakwater = 2L(L ₁ + L ₂ = 2L), where L is the wave length.

The present numerical model and experimental works of Koji and Mutsuo²² presented the wave height of regular waves for the purpose to compare the results at the front face of the refracted breakwater; this is seen in Figure 2. In comparison with Figure 2, it is noticed that the diffraction coefficient at the front face of breakwater agree well between the two results. It is observed that the present numerical model can be appropriately used for practical applications in order to predict the wave fields inside and outside while the trench is dredged within the marine environment’s vicinity. This is possible due to comparisons made between the results from the numerical performances and experimental model tests for regular wave height.

Figure 2.

Comparison between the wave diffractions of this study and those of experimental works at the front face of refracted breakwater.

Numerical results and analysis

Numerical examples are presented to investigate the influences of the trench at the front face of refracted breakwater with different incident wave angles and varying depths of the trench on the seabed. The layout of the refracted breakwater and the rectangular trench outside the breakwater are shown in Figure 1. Computer program has been developed based on the theory stated above and numerical simulations were performed with the following conditions: the water depth of the vicinity of breakwater is expressed as $d_{1} = 7 m$ ; the three kinds of water depth within the submarine trench region are $d_{2} = 1.5 d_{1} = 10.5 m$ , $d_{3} = 2 d_{1} = 14 m$ , and $d_{4} = 2.5 d_{1} = 17.5 m$ ; the refracted angle within the refracted breakwater is $β = 160 \circ$ ; and the obliquely incident wave angles $θ = 45 \circ$ , $θ = 90 \circ$ , and $θ = 120 \circ$ are for the regular wave, respectively. Figure 3 presents the results for the wave height obtained by the present numerical model with and without the trench at the front face of refracted breakwater.

Figure 3.

Wave height distributions at the front face of refracted breakwater with or without trench for obliquely incident wave angle: (a) $θ = 45 \circ$ , (b) $θ = 90 \circ$ , and (c) $θ = 120 \circ$ .

In the case where the obliquely incident wave angle is $θ = 45 \circ$ (Figure 3(a)), a slight reduction in wave height is shown at the left part of the refracted breakwater corner, whereas the four other cases with or without the trench show similar wave height distribution. It is because the incident wave angle meets in direction with the refracted part of the breakwater and the effect of the trench is not revealed.

When applying the root mean square (RMS) ratio shown in Table 1, for the cases where the dredging trench is $d_{2} = 1.5 d_{1} = 10.5 m$ , the wave height is increased by 23.2% compared to that without the trench. For the cases where $d_{3} = 2 d_{1} = 14 m$ and $d_{4} = 2.5 d_{1} = 17.5 m$ , the wave height is reduced by 11.9% and 19.3%, respectively. Table 1 presents the effect of added trench for reducing the wave height for varying depth levels.

Table 1.

The reduction ratio of wave height at the front face of refracted breakwater.

Incident wave angle	Depth of trench
Incident wave angle	$d_{2} = 1.5 d_{1}$	$d_{2} = 2 d_{1}$	$d_{2} = 2.5 d_{1}$
$θ = 45 \circ$	−23.2%	11.9%	19.3%
$θ = 90 \circ$	42.1%	35.1%	19.2%
$θ = 120 \circ$	42.3%	43.5%	36%

In the case where the obliquely incident wave angle is $θ = 90 \circ$ (Figure 3(b)), the wave height reduction is the largest at the vicinity of the corner side. This shows that the wave energy is concentrated in the corner of the refracted breakwater and can be subdued using the trench. Such act can reduce damage on the breakwater. The entire region exhibits a reduction in height as shown in Figure 3(b). This reduction is 42.9% less than the numerical results given for the case without the trench where $d_{2} = 1.5 d_{1}$ , 35.1% less for $d_{3} = 2 d_{1}$ and 19.2% less for $d_{4} = 2.5 d_{1}$ .

As for the case where the inclined wave angle is $θ = 120 \circ$ , the most significant reduction in wave amplitude occurs at the calculation point. The reasons are the incident wave angle is in normal direction with the refracted part of the breakwater and effect of the trench is remarkably revealed. The following estimations can be made in terms of wave height reduction due to the presence of the trench: 42.3% for $d_{2} = 1.5 d_{1}$ , 43.5% for $d_{3} = 2 d_{1}$ , and 36% for $d_{4} = 2.5 d_{1}$ .

As shown in Figure 3, among the three cases of trench depth when $d_{2} = 1.5 d_{1} = 10.5 m$ , the wave height reduction for the specific incident wave angle does not occur, whereas when $d_{4} = 2.5 d_{1}$ , the reduction occurs but the trouble to deepen the dredge depth exists. Therefore, as for $d_{3} = 2 d_{1}$ , it can be concluded that when the trench depth is dredged twice the measurement as the water depth, the wave height reduction effect can be significantly observed in the vicinity of the breakwater region.²¹

From these analyses, we see that when the trench is appropriately set, the overall reduction of wave height and wave energy can be observed at the vicinity of the breakwater.

Figures 4 and 5 present the contours of the wave height ratios near the breakwaters with and without the presence of the trench. The presence of the trench reduces both the wave amplitude and the wave energy within, around, and directly downstream of the trench. Based on the results, a dredging trench is deduced to provide an excellent means of protection from a wave attack. It seems that a discontinuity in water depth (dredging) weakens wave energy. The numerical simulation for wave diffraction has provided results that show the present model is useful in estimating the wave field at the vicinity of breakwater when a trench is dredged on the offshore seabed.

Figure 4.

Wave height contour plots within the vicinity of breakwater with or without trench (where the obliquely incident wave angle is $θ = 120 \circ$ ).

Figure 5.

Wave height contour plots within the vicinity of breakwater with or without trench (where the obliquely incident wave angle is $θ = 45 \circ$ ).

Figures 6 and 7 present three-dimensional diagrams of the wave height ratios near the breakwaters with and without the presence of the trench. As shown in the figures, the incident wave energy propagating from the offshore passes over the trench and the wave height is reduced by the discontinuity water depth. In addition, in the region between trench and the refracted breakwater, the diffracted wave height is low and thus shows that the trench is helpful for the safety of breakwater. Even afterwards, the reduced wave energy continuously propagates to the leeward of the breakwater and that region remains low. This contributes to the stillness of the fishery ports or harbors.

Figure 6.

Three-dimensional diagrams within the vicinity of breakwater with or without trench (where the obliquely incident wave angle is $θ = 120 \circ$ ).

Figure 7.

Three-dimensional diagrams within the vicinity of breakwater with or without trench (where the obliquely incident wave angle is $θ = 45 \circ$ ).

On the other hand, for the case where the trench is not installed, high wave height distribution is shown in the diagrams and this, as a result, affects the safety of the breakwater. In this case, the peak wave crest is distributed at the front face of the breakwater.

For the case where the trench is present, by the discontinuity of water depth, the wave is reflected toward the offshore direction resulting the wave crest between the trench and breakwater to be mild. Compared to the peak wave crest, this mild wave crest distributes the wave force which is concentrated at the front face of the breakwater, in a more effective manner for the safety of the breakwater.

Conclusion

The objective of this study is to predict the decreasing effects of scattered wave fields around the breakwater when the dredging trench exists at the offshore seabed. This study aims to address the composite wave interaction for the following three problems: dredging boundaries, depth discontinuity of trench, and breakwater boundaries. Boundary conditions are placed and applied to a 2D boundary integral model. The results obtained from the obliquely incident wave conditions are given to highlight the wave height distribution of the wave field near breakwater by the influence of the various trench depths. The results acquired from the present numerical model were compared with those from the previous experiments; the numerical simulations given show excellent consistency with the experimental data for wave height.

Through the present numerical simulations, the reduction in wave height and the weakness of wave energy around the refracted breakwater can be observed when the breakwater exhibits different incident wave angles due to the trench with varying depths at the offshore seabed.

The results from the present developed numerical model proved to provide accurate accounts of the scattered wave height of vicinity of breakwater systems when the trench is dredged on the offshore seabed. Thus, such model can be trusted to be used in the layout of breakwater planning, deepening or widening of a dredge, design applications of harbor, and redevelopment of harbor or port.

Footnotes

Academic Editor: Haitao Yu

Declaration of conflicting interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Funding

This research was supported by a grant (13RDRP-B066173) from Regional Development Research Program funded by Ministry of Land, Infrastructure, and Transport of Korean government.

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The wave energy scattering by interaction with the refracted breakwater and varying trench depth

Abstract

Keywords

Introduction

Theoretical development

Validation of the numerical model

Numerical results and analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References