1. Introduction
The coast is a site where several phenomena can appear and affect coastal structures, such as sea-level rise caused by climate change, erosion due to wave actions, and the decrease of fluvial sediment supply caused by the construction of dams, etc. Consequently, researchers, engineers, and scientists have shown that the reflection of swell is one of the convincing and relevant solutions to overcome problems of erosion, surges, marine submersions, and all phenomena to which these structures can be exposed. Therefore, the maximization of wave reflection has become extremely important in coastal engineering, and arouses the interest of many researchers in the literature to investigate the numerical, experimental, and theoretical methods that can study the reflection of wave–structure interactions.
Considerable research has been carried out in the past on different types of breakwaters. Nevertheless, the operating conditions of breakwaters are very challenging and, hence, there is still a need for additional research to fill in the gaps. A short account of what has been delivered up till now is given here. Dean [
1] studied the effect of the wave amplitudes on the reflection of surface waves by a submerged plane barrier. Takano [
2] evaluated the passage effect of waves propagating under a rectangular breakwater. Patarapanich [
3] studied the wave reflection and transmission by a submerged thin horizontal plate using the finite element method (FEM). Using the matched asymptotic method, Liu and Jiankng [
4] investigated the transmitted wave intensity through a submerged slit on a vertical barrier. On the other hand, using a parametric experimental design, Stamos et al. [
5] compared the reflection and transmission coefficients resulting from the interaction of waves with a variety of submerged water-filled breakwater models of hemi-cylindrical and rectangular shapes. Molin et al. [
6] carried out laboratory experiments to investigate the interaction of waves with a rigid vertical plate. Shortly afterwards, Lui et al. [
7] examined the Bragg reflections of water waves by multiple submerged semi-circular breakwaters.
Further, the submerged rectangular step is a structure mostly used as a breakwater to protect shorelines by diminishing the destructive effects of the wave actions, reducing the erosion of coasts, and protecting coastal structures from damage [
8]. Recently, the rectangular submerged breakwater has started to receive more attention compared to the traditional emerged structures, due to its attractive aesthetics and ability to allow water circulation as well as the passage of fish. Several experimental, analytical, and numerical studies have been devoted to studying the reflection and transmission of swells by this type of structure. Mei and Black [
9] studied the problem of scattering properties for bottom and surface obstacles using the variational method. Massel [
10] investigated the interaction of waves with an infinite- and finite-length rectangular submerged breakwater. Andrew et al. [
11] presented an experimental and numerical study based on the boundary element method (BEM) to investigate the propagation of waves over a submerged impermeable obstacle of a rectangular cross section. Recently, Szmidt [
12] studied numerically using the finite difference method (FDM), the interaction of waves with a rectangular breakwater fixed at the bottom of numerical wave tank (NWT), and estimated the efficiency of the breakwater in protecting sea shelf zones from open sea waves.
We propose to investigate, in this paper, the capabilities of the improved version of the meshless singular boundary method (ISBM) [
13,
14,
15] to analyze the reflection and transmission coefficients resulting from the interactions of regular waves with a rectangular breakwater sited at the bottom of a tank. The method is validated by studying the accuracy of the numerical results with respect to the number of boundary nodes and the location of the vertical boundaries of the computational domain for different immersion ratios (
h/d) and different relative lengths (
w/d) of the obstacle. Further, the analytical reflection and transmission coefficients within the plane wave model (see
Appendix A) are compared with the results of the numerical model and discussed for several wave and structural conditions. To improve the shortfalls of the analytical model, slight modifications are introduced to the analytical procedure, which is termed here the corrected analytical plane wave model. Finally, a general discussion is made to highlight the strengths and limitations of the corrected plane wave model.
This paper is divided into five main sections. After presenting the introduction of the work and stating the objective of this research in
Section 1, the formulation of the problem and the numerical method used in this work are presented in
Section 2 and
Section 3. In
Section 4, the error sensitivity indicator, and comparisons of the numerical and analytical results and analysis, are presented to study the interactions of regular waves with a rectangular breakwater sited at the bottom of a tank. Finally, some conclusions and perspectives are illustrated in
Section 5.
2. Formulation of the Problem
We consider, in this study, a submerged single impermeable rectangular step (breakwater) which is placed at the bottom, as shown in
Figure 1.
The idealized geometry of the two-dimensional (
2D) problem in a Cartesian system (
x-y) is shown in
Figure 2. Regular waves of small amplitude
a, period
T, and wavelength
L impinge from the left in water of depth
d. Assuming an irrotational flow and incompressible fluid motion, the problem is formulated using a velocity potential
where
Re denotes the real part,
is the time independent spatial velocity potential,
is the wave angular frequency, and
t is the time. The wave number
is the solution of the dispersion relation
, where
g is the gravitational acceleration. The wave field is totally specified if the two-dimensional velocity potential
is known.
The breakwater is described by the immersion ratio h/d and relative length (w/d), where d is the water depth in the absence of the obstacle, h is the water depth above the obstacle, and w is the length of the obstacle.
The total fluid domain is divided into three regions as shown in
Figure 2. Region I at (−∞) is the region where the waves are incoming (inflow), and region III at (+∞) is where the waves are transmitted (outflow). Region II is between regions I and III, and is delimited by the rigid (impermeable) walls of the breakwater (
,
, and
), the free surface boundary
, the seabed boundary
, and the radiation boundaries
and
, respectively, of the inflow and outflow regions. The spatial velocity potential
satisfies the following conditions:
where
n is the normal to the boundary pointing out of the flow region, and
denotes the total rigid (impermeable) boundary of the breakwater.
The radiation conditions at the inflow and outflow regions are expressed as
where
is the incident velocity potential.
The radiation conditions in the infinite strip problem are treated by transferring the far field potentials at two fictitious vertical boundaries at finite distances
and
, representing, respectively, the left boundary
and the right boundary
of the fluid domain. The analytical series at these boundaries are given by:
where
and
are unknown complex coefficients to be determined. The disturbances are guaranteed to be out-going waves only (see for example [
16,
17]). The incident velocity potential is defined as:
The special matching conditions at the interfaces
and
of the flow regions ensure a smooth transfer of the mass flow from one region to the next. Once the potentials
and
are calculated by satisfying the radiation boundary conditions of Equations (5) and (6), they are matched to those of Equations (7) and (8), then the unknown coefficients
and
are evaluated following the method of Yueh and Chuang [
18]:
where
.
The reflection and transmission coefficients (
R and
) are determined from the following expressions (see [
16,
17] and further details in
Appendix B):
3. Numerical Solution by the ISBM
For the numerical solution the total boundary of the whole computational domain is discretized as shown in
Figure 3 for the single breakwater.
In the ISBM, the nodal values of the potentials and their fluxes are expressed as linear combinations of fundamental solutions and their derivatives [
13,
14,
15],
where
are unknown coefficients to be determined,
and
are, respectively, the collocation points
and the source points
, and
N is the total number of points.
and
are the Dirichlet and Neumann values, and
is the normal at the collocation point
. The coefficients
and
are source intensity factors corresponding, respectively, to the fundamental solution and its derivative.
is the fundamental solution of the 2D Laplace equation. It depends only on the Euclidean distance
between the collocation points
and the source points
, i.e.,
, and is given together with its normal derivative as:
and
are the component of the normal at the collocation point
.
The coefficients
and
are the diagonal elements of the ISBM interpolation matrices. They arise when the collocation points and the source points coincide (
). Direct evaluation of these coefficients is unfeasible because of the singularities inherent in the fundamental solution and its derivative. In this study, the coefficients
are evaluated simply by the integration of the fundamental solution on line segments leading to a simple analytical expression as [
19,
20]:
For the coefficient
, a simple expression is derived by Gu [
14], using a regularization process of subtracting and adding-back to remove singularities:
where
and
are the half distances, respectively, between the collocations points
and
, and the source points
and
.
is the normal at the source point
.
The boundary conditions given by Equations (2)–(6) are satisfied by a linear combination of Equations (13) and (14). The discretization process leads to:
For nodes
(free surface boundary):
For nodes
(the radiation boundary at
):
For nodes
(the radiation boundary at
):
For nodes
and
(seabed and breakwater boundaries):
The resulting discretized Equations (19)–(22) are written in a more compact matrix form as:
where
N is the total number of nodes on the whole domain boundaries, e.g.,
, where
,
,
,
, and
are the number of nodes respectively on the boundaries
,
,
,
, and
. The algebraic system of equations expressed by Equation (23) is solved numerically using a Gaussian elimination algorithm to yield the vector of unknowns
. The potential and its derivative at the nodes are then computed using Equations (13) and (14).
5. Conclusions and Perspectives
In this research paper, the reflection and transmission coefficients during the interactions of regular wave-rectangular breakwater cited at the bottom are studied using the ISBM approach, the analytical approach within the plane wave model (
Appendix A), and the corrected plane wave model. Firstly, to verify the capability of the proposed NWT, the error sensitivity to the total number of boundary nodes, and to positions of fictitious vertical boundaries, for different immersion ratios (
h/d) and for different relative length (
w/d) of the obstacle, are deeply investigated. Further, the results of this work show that the minimum number of boundary nodes
N = 200 is required to ensure an accurate computation solution for narrow breakwaters, and
N = 600 for wider breakwaters. Furthermore, to ensure minimal values of errors, the quantity
is recommended to be in the range
for narrow breakwaters, and
for wider breakwaters.
Next, the improved version of the meshless singular boundary method (ISBM), the analytical approach within the plane wave model (
Appendix A), and the corrected plane wave model are compared to investigate the capacity of the plane wave and corrected plane wave models to study the reflection and transmission coefficients during interactions of regular wave-rectangular breakwater cited at the bottom of tank. For the sake of details, the plane wave model is efficient for small relative water depths
kd; whereas, by correcting the plane wave model by adding 5% to the relative length, and subtracting 4% from the immersion ratio, the corrected plane wave model appears to be in an acceptable agreement with the ISBM approach. Afterwards, the results show that the corrected plane wave model is successful in deeply analyzing the effects of relative length and immersion ratio on reflection and transmission coefficients. Then, it is recommended to use this approach compared to the ISBM method that is conditioned to the nodes number and the position of fictitious vertical boundaries. As perspective, we endeavor to study the wave-current–structure interactions using the Generating-Absorbing Boundary Conditions (GABCs) approach [
22] to meticulously study the reflection and transmission coefficients for different aspects of currents.