Hydroelastic Response to Oblique Wave Incidence on a Floating Plate with a Submerged Perforated Base

Abstract

A hydroelastic model is developed of a floating flexible structure in the presence of a submerged perforated base connected with mooring lines under oblique wave action. Using the velocity decomposition method, the analytical solution of the referred model is obtained in finite water depth. The convergence of the analytical solution for different oblique wave incidences is examined, and the present results of deflection amplitude are compared with experimental datasets and the numerical results available in the literature. The effects of oblique wave incidence, along with various design parameters on the reflection, transmission, and dissipation coefficients, as well as structural displacements, are analysed through hydroelastic analysis. Further, the effect of oblique incidence angle on the free oscillation hydroelastic waves in two wave modes is investigated by deriving the free motion velocity potential in a wave basin.

1. Introduction

Due to the rise in marine and human activities in offshore and coastal regions, there is considerable significance in the application of horizontal flexible floating and or submerged structures as breakwaters for protecting coastal infrastructures and offshore platforms. To meet the increasing demand for coastal protection and utilization of ocean space, there is a need to develop and analyse multi-use flexible structure systems that could be applied as floating breakwaters or land-filled artificial islands. Therefore, for practical engineering design, it is necessary to examine the hydrodynamic performance of flexible structures through hydroelastic analysis (see [1,2,3]). Hence, the interest in designing or modelling horizontal floating and submerged flexible porous structures have increased to investigate their effectiveness via hydroelastic analysis.

These types of horizontal structures are unique in nature, primarily because of their flexibility; displacement; and associated hydroelastic response, analysis, and design when compared with fixed structures. Constructions of these structures are cost-effective when the water depth is large as they are easy and fast to construct and can be removed or expanded with ease. These structures are protected from seismic shocks because the energy can be dissipated into the infinite ocean, and they do not damage the marine eco-system or obstruct the ocean current and are thus environmentally friendly. On the other hand, the effectiveness of these flexible porous structures is due to the fact that they are able to reflect, absorb, and dissipate wave energy. Moreover, the use of flexible porous structures has great advantages compared to rigid impermeable structures because the structural porosity helps in dissipating a large amount of wave energy, and the flexibility of the structure provide the additional feature of wave attenuation through structural displacement [4].

Commonly, the potential theory is used for the analysis of the linearized hydroelastic problem under the assumption of small wave amplitude. The classical thin-plate theory is frequently employed to analyse the dynamic response of floating structures (see [5,6]). On the other hand, the dynamic response of offshore platforms and wave loads on floating bridges were studied in [7]. The hydroelasticity modelling challenges associated with problems in various areas of engineering applications in offshore and arctic regions are reviewed in [8]. The mathematical theory and methods of hydroelasticity for solving problems associated with wave interaction with floating flexible structures are viewed as the major challenges to adequate and successful treatment of such complex models in the application of technology development.

In past studies, there have been several types of numerical methods successfully used in marine engineering to investigate the hydrodynamic effect of fluid-structure interaction problems. For example, the Finite Volume Method in [9], the Arbitrary Lagrangian-Eulerian Method in [10], particle-based methods, such as the Smoothed Particle Hydrodynamics in [11], coupled methods, such as coupled SPH-DEM in [12], coupled SPH-FEM in [13], and ANSYS^® AQWA (henceforth called only AQWA for simplicity), which is a BEM code [14]. On the other hand, the hydroelastic analysis of various geometries of very large floating structures (VLFS) was studied based on different numerical methodologies. For instance, the hydroelastic analysis of VLFS by attaching an annular plate below VLFS using the Rayleigh–Ritz technique was investigated in [15]. The hydroelastic response of VLFS using vertical elastic mooring lines was studied in [16] based on the hybrid finite element–boundary element method. Further, a Finite Element-Multi-Domain Boundary Element method was presented for the hydroelastic analysis of VLFSs with perforated plates attached at the fore or aft under linear wave theory in [17]. A hydroelastic model based on the multi-mode expansion and the Finite Element Method (FEM) was developed in 2D for VLFS in [18], along with a coupled-mode model in 3D over the variable bottom (see [19]). Based on the BIEM approach, the effect of mooring lines on a floating flexible plate was analysed in [20]. The hydroelastic responses of a submerged horizontal porous plate attached at the front of a VLFS of rectangular type were analysed in [21] based on the matched eigenfunction expansion method (MEFEM).

Another interesting aspect of this class of problem is the analysis of wave loads, displacements, and the moment of the structure under the action of waves. The wave-induced loads on the deck and beam elements were measured based on physical model tests of exposed piers/jetties/bridges in [22]. A finite-element analysis (FEA) code, LS-DYNA, is used to perform the CFD analysis and compute the wave loading on coastal bridges in [23]. A large-scale hydrodynamic experiment of tsunami wave interaction with an I-girder bridge with cross frames was presented to analyse the impact of tsunamis on bridge inundation mechanisms as well as the associated connection forces between the super- and sub-structures of these bridges [24]. The significant effect of the trapped air on the pressures, total forces, and the response of coastal bridges were studied in [25] based on experiments. Further, the forces on a bridge superstructure in the horizontal and vertical directions based on experimental and numerical hydrodynamic models were investigated to determine the effect of variations in bridge elevation and wave height on the magnitude of the horizontal and vertical forces and overturning moment [26].

There has been considerable progress in understanding the behaviour of submerged horizontal porous structures and their effect on floating flexible structures based on MEFEM. The interaction of oblique monochromatic waves with a submerged horizontal porous plate in the framework of two-dimensional potential flow was inspected in [27] based on MEFEM, and reflection and transmission coefficients were further studied using the Boundary Element Method (BEM) in [28]. In [29], a time-domain hybrid FEM-BEM was presented to investigate the interaction between oblique irregular waves and a very large floating structure (VLFS) edged with dual inclined perforated plates.

The hydroelastic response of a mat-like rectangular VLFS with dual horizontal inclined porous anti-motion plates using BEM was investigated in [30]. The hydroelastic problem of a pontoon-type VLFS connected with porous plates and non-porous plates was studied in [31], both numerically and experimentally. Under velocity decomposition, an analytical model of surface gravity waves with a submerged flexible membrane in the presence of a floating elastic plate connected with mooring lines was presented in [32]. Further, a 3D analytical hydroelastic model of a floating and a submerged flexible plate was developed (see [33]) to study the effect of mooring lines and modes of oscillation on a floating elastic plate in the presence of a submerged fixed membrane in the water of finite depth (FD) under ocean wave action.

Therefore, here, a mathematical model of a floating horizontal flexible structure in the presence of a submerged perforated structure connected with mooring lines of finite dimensions under oblique waves is developed to analyse the effect of oblique incident angle in FD.

The new contributions of the present physical model compared with [34] are the reduced wave equation, boundary conditions, and solutions, along with the comparison of results of deflection amplitudes with existing published experimental datasets and the numerical BEM-FEM model. Further, the effect of oblique wave incidence angle along with different design parameters on the moored floating flexible structure with a moored submerged porous structure is investigated by analysing the reflection, transmission, and dissipation coefficients as well as plate displacements in different cases via hydroelastic analysis. In addition, the effect of oblique incidence angle on the free oscillation hydroelastic waves in a wave tank is investigated in two wave modes.

2. Model Definition

A similar model in two dimensions was analysed in [34]. Here, the effect of oblique incident angle on the reflection, transmission, and dissipation coefficients and plate displacements are analysed in three dimensions, presenting significant contributions towards formulation and solutions, and the results are discussions.

The model is defined under the hypothesis of linear wave theory and the Cartesian variables $(x,y,z)$, where the $x-z$ plane is considered as the still position of the water surface and the y-axis is pointing positively downward. A horizontal floating and submerged porous plate with a length of $2L$ occupies $-L<x<L$ $0<z<\infty$ and is placed at $y=0$ (floating plate), and a porous plate is situated at $y=d$ that also occupies $-L<x<L$ and $0<z<\infty$ with the bottom bed $y=h$ FD. For practical interest, both plates are connected with mooring lines with stiffnesses $q_{fj}$ and $q_{sj}$ at $x=\pm L$ for $j=1,2$, and they are extended along the z-direction at $0<z<\infty$. Structural dimensions and problem definitions are presented in Figure 1.

Therefore, the entire fluid region is bifurcated as:

$$\mathrm{Fluid} \mathrm{domain}=\{\begin{array}{l}-\infty <x<-L,L<x<\infty ,y\in (0,h), 0<z<\infty (\mathrm{Open} \mathrm{water} \mathrm{region}),\\ -L<x<L,y\in (0,d)\cup (d,h), 0<z<\infty (\mathrm{Plate} \mathrm{covered} \mathrm{region}).\end{array}$$

(1)

It is assumed that a progressive wave is obliquely incident with an angle, θ, on the floating and submerged porous plate along the x-axis. Further, the fluid is modelled as inviscid and incompressible, and the motion is irrotational.

It should be mentioned that there are many areas in applied mathematics and engineering, especially in fluid flows and elastic plate/membranes, which leads to the solution of the Laplace equation in two/three dimensions. The appropriate choice of boundary conditions is of paramount importance in the selection of appropriate mathematical tools to solve the problem of interest. Often, the three-dimensional Laplace equation is further simplified to $\mathrm{\Phi }(x,y,z;t)=\mathrm{Re}\{\varphi (x,y)e^{i(\gamma z-\omega t)}\}$, which reduces the three-dimensional Laplace equation to a two-dimensional elliptic PDE so-called reduced wave equation. Moreover, the z-component is absorbed by the term $\gamma$, which is associated with the z-component of the wavenumber ${\beta }_0$, associated with the incident waves in the open water region.

Therefore, the linear wave theory yields to the velocity potential, $\mathrm{\Phi }(x,y,z;t)=\mathrm{Re}\{\varphi (x,y)e^{i(\gamma z-\omega t)}\}$, for a fluid motion that is simple and harmonic in time, with an angular frequency, $\omega$, where $i=\sqrt{-1}$, $\gamma ={\beta }_0\sin \theta$ is the z-component of the wavenumber ${\beta }_0$ associated with the incident waves in open water regions, and $\varphi (x,y)$ refers to the spatial velocity potential and satisfies the reduced wave equation:

$$({\nabla }_{xy}^2-{\gamma }^2)\varphi =0, \mathrm{in} \mathrm{the} \mathrm{fluid} \mathrm{domain}$$

(2)

where ${\nabla }_{xy}^2={\partial }^2/\partial x^2+{\partial }^2/\partial y^2$.

The linearized free surface at $y=0$ is derived by combining the linearized kinematic and dynamic conditions as:

$${\varphi }_y+\tau \varphi =0,-\infty <x<-L,L<x<\infty \mathrm{with} \tau ={\omega }^2/g$$

(3)

The usual non-flow condition in water of FD at $y=h$ can be read as:

$${\varphi }_y=0,-\infty <x<\infty$$

(4)

Now, the boundary conditions on $y=0$ give (as in [34]):

$${\kappa }_1\frac{{\partial }^5{\varphi }_1}{\partial y^5}-{\epsilon }_1\frac{{\partial }^3{\varphi }_1}{\partial y^3}+\frac{\partial {\varphi }_1}{\partial y}+\tau {\varphi }_1=0 \mathrm{on} y=0 \mathrm{for} -L<x<L,z<\infty$$

(5)

The boundary condition on $y=d$ are given by:

$${\varphi }_{2y}={\varphi }_{1y}=-i\omega w_2+i\sigma ({\varphi }_2-{\varphi }_1) \mathrm{on} y=d$$

(6)

$$\begin{gathered} \left({\kappa }_2\frac{{\partial }^5{\varphi }_2}{\partial y^5}-{\epsilon }_2\frac{{\partial }^3{\varphi }_2}{\partial y^3}-m_{p2}\frac{\partial {\varphi }_2}{\partial y}+\tau {\varphi }_2\right)+i\sigma \left({\kappa }_2\frac{{\partial }^4}{\partial y^4}-{\epsilon }_2\frac{{\partial }^2}{\partial y^2}-m_{p2}\frac{\partial }{\partial y}\right)({\varphi }_2-{\varphi }_1)-\tau {\varphi }_1=0, \\ y=d,-L<x<L,z<\infty \end{gathered}$$

(7)

where ${\kappa }_j=E_j{I}_j/(\rho g-m_{pj}{\omega }^2)$, ${\epsilon }_j=N_j/(\rho g-m_{pj}{\omega }^2)$, and $\tau =\rho {\omega }^2/(\rho g-m_{pj}{\omega }^2)$. Further, $m_{p_j}={\rho }_{pj}{a}_j$ is the mass per unit length and $a_j$ is the thickness of the plate. In addition,$E_j$, $I_j=a_j^3/12(1-{\upsilon }^2)$, and $\upsilon$ is the Poisson’s ratio of each plate. In addition, $N_j$ is the uniform compressive force ($N_{jx}=N_{jz}=N_j$, $j=1,2$ where $N_1$ and $N_2$ are the uniform compressive forces acting on the plates). Further, $\sigma$ is the porous-effect parameter of the submerged flexible porous plate where $\sigma ={\sigma }_r+i{\sigma }_i$ is the complex porous effect parameter, as in [35], where ${\sigma }_r$ is the resistance force coefficient and ${\sigma }_i$ is the inertial force coefficient. The subscripts t and y refer to the partial derivatives w. r. t. t and y, respectively. Further, $w_2(x,z;t)$ denotes the submerged porous plate deflection. ${\varphi }_1$ and ${\varphi }_2$ denote the velocity potentials for the region $-L<x<L$, $0<y<d$, $0<z<\infty$ and $-L<x<L$, $d<y<h$, $0<z<\infty$, respectively. Further, $w_2$, defined in Equation (6), is the deflection of the submerged porous plate and is assumed by the following form:

$w_2(x,z;t)=\mathrm{Re}\{w_{20}(x)e^{i(\gamma z-\omega t)}\}$.

The moored edge conditions at $x=\pm L, y=0$ of the floating flexible plate give:

$$E_1{I}_1\left({\partial }_{xx}-\upsilon {\gamma }^2\right){\varphi }_y=0$$

(8)

$$\left[E_1{I}_1\left({\partial }_{xx}-(2-\upsilon ){\gamma }^2\right){\partial }_x+N_1{\partial }_x\right]{\varphi }_y=q_{fj}{\varphi }_y$$

(9)

Additionally, the moored edge conditions at $x=\pm L, y=d$ provide:

$$E_2{I}_2({\partial }_{xx}-\upsilon {\gamma }^2){\varphi }_y=0$$

(10)

$$\left[E_2{I}_2\left({\partial }_{xx}-(2-\upsilon ){\gamma }^2\right){\partial }_x+N_2{\partial }_x\right]{\varphi }_y=q_{sj}{\varphi }_y$$

(11)

Further, the following continuity conditions at $(0, r)$ with $r=0,d$ are necessary to solve this referred physical problem:

$${{\varphi }_y|}_{(0^{+},r)}={{\varphi }_y|}_{(0^{-},r)},{{\varphi }_{xy}|}_{(0^{+},r)}={{\varphi }_{xy}|}_{(0^{-},r)}$$

(12)

$$\begin{array}{l}{E_j{I}_j\left({\partial }_{xx}+\upsilon {\partial }_{zz}\right){\varphi }_y|}_{(0^{+},r)}={E_j{I}_j\left({\partial }_{xx}+\upsilon {\partial }_{zz}\right){\varphi }_y|}_{(0^{+},r)},\\ {\left[{\partial }_x[E_j{I}_j\{{\partial }_{xx}+(2-\upsilon ){\partial }_{zz}\}+N_j]{\varphi }_y\right]|}_{(0^{+},r)}={\left[{\partial }_x[E_j{I}_j\{{\partial }_{xx}+(2-\upsilon ){\partial }_{zz}\}+N_j]{\varphi }_y\right]|}_{(0^{-},r)}.\end{array}\}$$

(13)

The continuity of pressure and velocity at $x=0^{\pm },L^{\pm }$ give:

$$\varphi (0^{+},y)=\varphi (0^{-},y),\varphi (L^{+},y)=\varphi (L^{-},y)$$

(14)

$${\varphi }_x(0^{+},y)={\varphi }_x(0^{-},y),{\varphi }_x(L^{+},y)={\varphi }_x(L^{-},y)$$

(15)

In the end, the far-field radiation condition related to the incident, reflected, ane transmitted wave amplitudes, $I_0,R_0,\mathrm{and} T_0$, respectively give:

$$\varphi (x,y)=\{\begin{array}{l}\left(I_0{e}^{-i{\mu }_{0}x}+R_0{e}^{i{\mu }_{0}x}\right)\mathrm{\Omega }(y) \mathrm{as} x\to \infty ,\\ T_0{e}^{-i{\mu }_{0}x}\mathrm{\Omega }(y) \mathrm{as} x\to -\infty \end{array}$$

(16)

where $\mathrm{\Omega }(y)=\cosh {\beta }_0(h-y)/\cosh {\beta }_{0}h$ with ${\mu }_0^2={\beta }_0^2-{\gamma }^2$ and ${\beta }_0$ is the real root and satisfies the gravity wave dispersion relation ${\omega }^2=g{\beta }_0\tanh {\beta }_{0}h$ in the open water region.

2.1. Solution

By considering the geometrical symmetry of the physical problem regarding x = 0, the velocity potentials associated with the physical problems are rewritten as the sum of the symmetric and antisymmetric potentials for simplification, which can be handled easily for the solution at hand. Hence, the original problem is split into two simpler problems in the semi-infinite region along the $x$-axis of $0<z<\infty$. Here, to derive the system of equations, the orthogonal relation associated with the eigenfunctions in the open water region is used. Hence, it is sufficient to consider only the region $0<x<\infty$ of $0<z<\infty$ because the solution can be extended into the whole fluid domain using symmetric relations. Proceeding similarly as in [2], the split potentials, $\mathbb{L}(x,y)$ and $\widehat{\mathbb{L}}(x,y)$, fulfil Equations (1)–(6), and can be written as:

$$\mathbb{L}(x,y)=\{\begin{array}{l}{\displaystyle \sum _n^{}{a}_n{\mathrm{\Psi }}_n^S(x,y)}, 0<x<L, \mathrm{for} n=0,I,II,III,1,2,\dots \\ I_0{\mathrm{\Gamma }}_0(x,y)+{\displaystyle \sum _n^{}{b}_n{\mathrm{\Gamma }}_n(x,y)}, L<x<\infty , \mathrm{for} n=0,1,2,\dots \end{array}$$

(17)

$$\widehat{\mathbb{L}}(x,y)=\{\begin{array}{l}{\displaystyle \sum _n^{}{\overline{a}}_n{\mathrm{\Psi }}_n^S(x,y)} \mathrm{for} 0<x<L, \mathrm{for} n=0,I,II,III,1,2,\dots \\ I_0{\mathrm{\Gamma }}_0(x,y)+{\displaystyle \sum _n^{}{\overline{b}}_n{\mathrm{\Gamma }}_n(x,y)} \mathrm{for} L<x<\infty , \mathrm{for} n=0,1,2,\dots \end{array}$$

(18)

where ${\mathrm{\Psi }}^S(x,y)=J_n^S(x)f_n(y)$, $J_n^S(x)=(\cosh {\alpha }_{n}x)/(\cosh {\alpha }_{n}L)$, ${\mathrm{\Psi }}^A(x,y)=J_n^A(x)f_n(y)$, $J_n^A(x)=(\sinh {\alpha }_{n}x)/(\sinh {\alpha }_{n}L)$, ${\mathrm{\Gamma }}_0(x,y)=e^{-i{\mu }_0(x-L)}{\phi }_0(y)$, ${\mathrm{\Gamma }}_n(x,y)=e^{i{\mu }_n(x-L)}{\phi }_n(y)$,

$$f_n({\alpha }_n,y)=\{\begin{array}{l}\frac{Q_1({\alpha }_n,y)\mathrm{csch}{\alpha }_{n}d}{N_1({\alpha }_n,d)}, y\in (0,d),\\ -i\{\tau +i\sigma ({\kappa }_2{\alpha }_n^4-{\epsilon }_2{\alpha }_n^2)\}\frac{\cosh {\alpha }_n(h-y)}{\cosh {\alpha }_{n}H}, d<y<h,\end{array}$$

(19)

$$Q_1({\alpha }_n,y)=\{\tau +i\sigma ({\kappa }_2{\alpha }_n^4-{\epsilon }_2{\alpha }_n^2)\}\{{\alpha }_n({\kappa }_1{\alpha }_n^4-{\epsilon }_1{\alpha }_n^2+1)\cosh {\alpha }_{n}y-\tau \sinh {\alpha }_{n}y\}$$

(20)

where ${\alpha }_n^2={\beta }_n^2-{\gamma }^2$ and ${\alpha }_n$s are the zeros of the dispersion relation:

$$\begin{array}{l}Z(\alpha )={\tau }^2(1+\coth \alpha d\coth \alpha H)-\tau [\alpha ({\kappa }_1{\alpha }^4-{\epsilon }_1{\alpha }^2+1)\coth \alpha H\\ +\{\alpha ({\kappa }_2{\alpha }^4-{\epsilon }_2{\alpha }^2)+\alpha ({\kappa }_1{\alpha }^4-{\epsilon }_1{\alpha }^2+1)\}\coth \alpha d\\ -i\sigma ({\kappa }_2{\alpha }^4-{\epsilon }_2{\alpha }^2)(1+\coth \alpha d\coth \alpha H)]+{\alpha }^2({\kappa }_1{\alpha }^4-{\epsilon }_1{\alpha }^2+1)({\kappa }_2{\alpha }^4-{\epsilon }_2{\alpha }^2)\\ -i\sigma \alpha ({\kappa }_2{\alpha }^4-{\epsilon }_2{\alpha }^2)({\kappa }_1{\alpha }^4-{\epsilon }_1{\alpha }^2+1)(\coth \alpha d+\coth \alpha H),\end{array}$$

(21)

where $\tau ={\omega }^2/g$ and $H=h-d$; if $\gamma =0$ in Equation (16), then the truncated result will be the same as in [34]. All roots of the dispersion relation (21) are complex (for details, see [34] (below Equation (19)). It should be mentioned that, assuming the practical interest of the problem, only four roots are taken into account for the boundedness of the solution. The normalized vertical eigenfunction ${\phi }_n(y)$ is obtained by replacing ${\varphi }_n(y)$ in Equation (49) of [34].

2.2. Boundary Conditions in Connection with $\mathbb{L}(x,y)$ and $\widehat{\mathbb{L}}(x,y)$

Boundary conditions (7)–(9) need to be expressed in connection with $\mathbb{L}$ and $\widehat{\mathbb{L}}$ to determine the constants $a_n,b_n,{\overline{a}}_n, \mathrm{and} {\overline{b}}_n$ associated with Equations (10) and (11). Hence, conditions (8)–(11) can be reformulated as:

$$E_1{I}_1\left({\partial }_{xx}-\upsilon {\gamma }^2\right)\frac{\partial \mathbb{L}(L,0)}{\partial y}=0$$

(22)

$$\begin{gathered} {\partial }_x[E_1{I}_1\{{\partial }_{xx}-(2-\upsilon ){\gamma }^2\}+N_1]\frac{\partial \widehat{\mathbb{L}}(L,0)}{\partial y}=q_f\frac{{\partial }^2\widehat{\mathbb{L}}(L,0)}{\partial x\partial y} \\ {E}_2{I}_2\left({\partial }_{xx}-\upsilon {\gamma }^2\right)\frac{\partial \mathbb{L}(L,d)}{\partial y}=0 \end{gathered}$$

(23)

$${\partial }_x[E_2{I}_2\{{\partial }_{xx}-(2-\upsilon ){\gamma }^2\}+N_2]\frac{\partial \widehat{\mathbb{L}}(L,d)}{\partial y}=q_s\frac{{\partial }^2\widehat{\mathbb{L}}(L,d)}{\partial x\partial y}$$

(24)

where $\upsilon$ is the Poisson’s ratios of floating and submerged flexible plates.

Further, the continuity conditions (12) and (13) at (0, r) in terms $\mathbb{L}(x,y)$ and $\widehat{\mathbb{L}}(x,y)$ can be expressed as:

$${\mathbb{L}}_y(0,r)=0, {\mathbb{L}}_{xy}(0,r)=0$$

(25)

$$E_j{I}_j\left({\partial }_{xx}-\upsilon {\gamma }^2\right){\widehat{\mathbb{L}}}_y(0,r)=0$$

(26)

$${\partial }_x[E_j{I}_j\{{\partial }_{xx}-(2-\upsilon ){\gamma }^2\}+N_j]{\widehat{\mathbb{L}}}_y(0,r)=0$$

(27)

The continuity conditions (14) and (15) give:

$$\mathbb{L}(x,y)=0, \mathrm{at} x=L$$

(28)

$$\frac{\partial \widehat{\mathbb{L}}(x,y)}{\partial x}=0. \mathrm{at} x=L.$$

(29)

In the end, the condition as $x\to \infty$ satisfy:

$$\mathbb{L}(x,y)\approx \left(I_0{e}^{-i{\mu }_{0}x}+b_0{e}^{i{\mu }_{0}x}\right){\phi }_0(y)$$

(30)

$$\widehat{\mathbb{L}}(x,y)\approx \left(I_0{e}^{-i{\mu }_{0}x}+{\overline{b}}_0{e}^{i{\mu }_{m0}x}\right){\phi }_0(y)$$

(31)

where $b_0=R_0-T_0$ and ${\overline{b}}_0=R_0+T_0$ with $R_0$ and $T_0$ is the same as in Equation (16). The equation system presented in Appendix A can be solved numerically to determine $a_n,b_n,{\overline{a}}_n, \mathrm{and} {\overline{b}}_n$. Then, the reflection, transmission, and dissipation coefficients can be calculated along: $C_r=\left|(b_0+{\overline{b}}_0)/2I_0\right|$, $C_t=\left|(b_0-{\overline{b}}_0)/2I_0\right|$, and $C_e=1-C_r-C_t$, respectively.

To understand the effect of oblique incidence angle for different design parameters on the moored submerged flexible porous plate in the presence of a floating flexible plate, various results on the C_r, C_t, and C_e, as well as the submerged porous plate displacements, are analysed. From now on, all numerical results are performed by taking into account the same length, $2L$, for the floating and submerged plates; $\rho =1025{\mathrm{kgm}}^{-3}$; $g=9.8{\mathrm{ms}}^{-1}$; mooring stiffness, $q_f=q_s={10}^2{\mathrm{Nm}}^{-1}$; Young’s modulus, $E_1=0.47\times {10}^9{\mathrm{Nm}}^2$ and $E_2=0.47\times {10}^7{\mathrm{Nm}}^2$; compressive force on the floating plate, $N_1=0.5{(E_1{I}_1\rho g)}^{1/2}$; the submerged porous plate, $N_2=0.2{(E_2{I}_2\rho g)}^{1/2}$; $d/h=0.25$; $L/h=2.5$; $\sigma =1.0+0.5i$; and plate thickness, $a_1=a_2=0.1\mathrm{m}$, if not otherwise specified.

MATLAB R2016b (MATLAB 9.1, source: mathworks.com/trademarks), 64-bit (win64) was used to perform calculations in a very quick manner. All numerical computations of the analytical expressions were performed on a desktop machine with Intel ^® core i7–7700 CPU with a4.20 GHz processor and 16 GB of RAM, 3601 Mhz, 4 Core(s), and 8 Intel Processor(s) ). The data was written onto an SSD disk and, on average, each case took roughly 8–10 min to complete. Further, using the present solutions, MATLAB codes were developed to simulate the C_r, C_t, and C_e as well as the plate displacements.

3. Numerical Results and Discussions

It should be noted that to avoid repetition, the results analysis of the C_r, C_t, and C_e. for different design parameters versus non-dimensional wavelengths are deferred here (see [34]). Here, the new contribution is the analysis of the effect of oblique incidence angle on the reflection, transmission, and dissipation coefficients, and in addition, the structural displacements are presented.

Recently, the convergence of the C_r and C_e of a similar problem in two-dimensions was studied in [34] for different mooring stiffness and compressive force. Here, the convergence of the C_r and C_t of the series solution is checked for different oblique incidence angles with $d/h=0.6$ and wave period T = 5 sec.

Table 1. Convergence analysis.

Parameter	$\mathit{N}$	Cr	Ct
0o	2	0.776671	0.122774
	3	0.776668	0.122773
	6	0.776669	0.122774
30o	2	0.782145	0.137840
	3	0.775545	0.125941
	6	0.775542	0.125940
45o	2	0.785870	0.139198
	3	0.774964	0.129311
	6	0.774961	0.129310
75o	2	0.801884	0.140470
	3	0.779231	0.135731
	6	0.779228	0.135730

demonstrates the values of C_r and C_t converged to N ≥ 6 , and hence for computational accuracy, $\mathrm{N}=6$ is confined to compute C_r, C_t, and C_e.

In Figure 2, the comparison of deflection amplitude $(\mathrm{m})$ of the floating flexible plate between the present analytical against experimental datasets (see [30], Figure 8a) and the numerical FEM-BEM results (see [30], Figure 9c) versus non-dimensional wavenumber, ${\beta }_{o}L$, are plotted. The trends of the deflection amplitude are well captured and for smaller values of wavenumber, the data points are well in agreement between the two models. However, some data points are not matched well with the analytical results for higher wavenumbers.

It should be mentioned that the interaction of a VLFS with a dual submerged horizontal porous plate is investigated in [30], where the ends of the VLFS were not moored. Here, the smaller values of mooring stiffness, $q_f=q_s={10}^2{\mathrm{Nm}}^{-1}$, and the critical value $N_2=0.5{(E_2{I}_2\rho g)}^{1/2}$ in Figure 2 is chosen to approach the free edge conditions and less compressive force. Therefore, the behaviour of the results between the model (see [30]) and the present one is deemed to be similar.

Comparison results showed that the trend of the deflection amplitude between the experimental and the numerical FEM-BEM (see [30]) with the present analytical results are all well captured. However, a very few experimental data points in the deflection amplitude are a little far from the model (see [30]) with a higher wavenumber. These deviations may be explained by the present solution associated with the two propagating wave modes along with compressive force in the flexible plates. Additionally, the effects of complex wavenumber and the compressive force considered in the present model can take part in the behaviour of the deflection pattern of the flexible plate, while these effects were not considered in [30]. Therefore, it is suspected that the present model reproduces a little lower deflection amplitude for higher non-dimensional wavenumber that the model (as in [30]) cannot reproduce.

3.1. Effect of Oblique Incidence Angles on the Reflection, Transmission, and Dissipation Coefficients

Figure 3 displays the effect of the $C_r$, $C_e$, and $C_t$ versus oblique angle θ (deg.) for different mooring stiffness, $q_f$, of the floating flexible plate with $q_s={10}^2{\mathrm{Nm}}^{-1}$. The reflection coefficients, $C_r$, will be higher as the values of $q_f$ increase, whilst the dissipation coefficients, $C_e$ and $C_t$, decreases. However, for θ = 90°, it is seen that all incident waves neither reflected nor dissipated. This is due to the fact that both the floating and the submerged porous plates are considered to be of infinite length in the z-direction. Therefore, when the incident wave is perpendicular to the plate system, it propagates along the plate system in the z-axis direction only, and no there are no waves in the x-axis direction.

Figure 4 shows the effect of $N_1$ on the reflection, transmission, and dissipation coefficients against $\theta$ with $N_2=0.2\sqrt{E_2{I}_2\rho g}$. As observed, the compressive force increases as $C_r$ increases, while $C_e$ and $C_t$ become lower. This may be because, with an increase in the compressive force on the floating flexible plate, it becomes stiffer with a certain value of mooring stiffness in the submerged porous plate, which leads to higher reflection and smaller dissipation and transmission.

In Figure 5, the comparison results of the $C_r$,$C_t$ and $C_e$ for various $d/h$ versus angles $\theta$ are simulated. More importantly, the dissipation coefficient increases with an increase in the values of depth of submergence d/h, while the effect of transmission coefficient is negligible, as seen in Figure 5, which is due to a large portion of the wave energy being absorbed by the submerged porous plate.

Figure 6 gives the variations $C_r$,$C_t$, and $C_e$ against θ on several structural lengths, $L/h$. It is observed that as the plate length increases, the $C_r$ becomes lower and wave energy absorption over the submerged structure increases. This is due to the sizeableness of the flexibility and the pores in the plate.

Figure 7 demonstrates the effect of the porous effect parameter on the reflection and dissipation coefficients $C_r$ and $C_e$ versus θ. It is observed that as the values of the porous parameter in the submerged flexible plate increase, the reflection coefficients, $C_r$, decreases, while the dissipation coefficients increase only for $\theta >{30}^o$. However, for smaller angles,$\theta <{30}^o$, the trend of C_e is reversed to that of $\theta >{30}^o$, which is due to the change of phase of waves when the resistance force coefficient and inertial force coefficient of the porous-effect parameter are different.

It may be noted that, in Figure 3, Figure 4, Figure 5 and Figure 6, the trend of the $C_r$,$C_t$, and $C_e$ is significantly dependent on the value of mooring stiffness, compressive force, submergence depth, and plate length. For small wave angles (θ < 55°), the reflection coefficient becomes higher for higher mooring stiffness, submergence depth, and compressive force, while the $C_t$ and $C_e$ follow the reverse pattern. However, for θ = 90°, all incident waves neither reflected nor dissipated.

3.2. Structural Displacements via Hydroelastic Analysis

Figure 8 depicts the displacements, $w_2(\mathrm{m})$, of the moored submerged flexible porous plate for various oblique incidence angles: $\theta$ for $\sigma =0.5i$ versus ${\beta }_{0}h$ and plate length $L(\mathrm{m})$. From Figure 8a–d, it can be seen that, for higher oblique incidence angle, $\theta$, the displacement of the submerged porous plate decreases, and only a single peak occurs at the edges of the plate.

Figure 9 demonstrates the displacements, $w_2(\mathrm{m})$, for different submergence depths with an incident angle versus non-dimensional wavenumber and plate length. From Figure 9a–d, it can be observed that, for higher submergence depth, the displacement amplitude of the submerged flexible plate becomes lower. This is due to less interaction with the floating plate and absorption of wave energy over the submerged porous plate.

Figure 10 simulates the displacements, $w_2(\mathrm{m})$, for various compressive forces, $N_1$, with $\sigma =0.5i$ and $\theta =60 (\mathrm{deg}.)$ versus non-dimensional wavenumber, ${\beta }_{0}h$, and plate length, $L(\mathrm{m})$. In Figure 10a–d, for higher values of $N_1$, the deflections become smaller, which is due to the increased wave energy reflected as a result of less wave energy being absorbed by the submerged porous plate, which leads to lower deflection. Although the number of crests is the same in each case (a–d), the displacement amplitudes decrease as the values of compressive force become higher.

Figure 11 displays the submerged porous plate displacements for different mooring stiffness: (a) $q_s={10}^1{\mathrm{Nm}}^{-1}$, (b) $q_s={10}^2{\mathrm{Nm}}^{-1}$, (c) $q_s={10}^3{\mathrm{Nm}}^{-1}$, and (d) $q_s={10}^5{\mathrm{Nm}}^{-1}$ with $d/h=0.25$. In Figure 11a–d, it can be clearly seen that the plate displacement decreases with an increase in the values of mooring stiffness. This is due to the fact that, as the stiffness of the mooring line increases, the flexibility of the plate becomes harder as a result of the decrease in the displacements of the flexible plate.

It should be noted that there is only one single peak in each Figure (Figure 8, Figure 9, Figure 10 and Figure 11) in the deflections being observed, which is due to the phase change of the incident and reflected waves, leading to constructive/destructive interference of the waves at the edge of the structure.

In the next section, the free oscillation hydroelastic waves in a wave tank of floating flexible plate in the presence of a submerged porous plate will be analysed in a specific case in two wave modes. In the case of surface wave interaction with a floating flexible plate in the presence of a horizontal flexible porous plate, two types of wave mode exist due to the interaction of waves with the floating plate and submerged flexible porous plate. One corresponds to the floating flexural mode (FFM) and another one corresponds to the submerged flexural mode (SFM). Further, the waves in the FFM are generated due to the presence of the floating flexible plate and the waves in SFM are generated because of the submerged flexible porous plate.

4. Effect of Oblique Incidence Angle on the Hydroelastic Waves in a Wave Tank in FFM and SFM

The role of different design parameters on the hydroelastic waves in FFM and SFM in a two-dimensional closed wave tank was inspected in [34]. Here, the effects of oblique incidence angle on the phase and group velocities, dispersion curves, and period of oscillations in FFM and SFM are investigated.

Consider a basin bounded by vertical planes, $x=0,l$, and a horizontal flexible plate floating at the free surface with a porous plate submerged below the floating plate; a detailed sketch of the arrangements is shown in Figure 12. Further, the basin length of $l(m)$ and $\gamma ={\beta }_0\sin \theta$ present the z-component of the wavenumber, ${\beta }_0$, related to the plane progressive waves in the open water region, with an incidence angle of $\theta$. It is assumed that the fluid characteristics and structural response is the same as discussed in Section 2.

The velocity potential, $\varphi (x,y)$, associated with the free motion can be expressed by satisfying Equations (1) and (3), boundary conditions (4)–(6) with $0<x<l$, and ${(\partial \varphi /\partial x)|}_{x=0,l}=0$ as:

$$\mathrm{\Phi }(x,y;t)={\displaystyle \sum _{n=1}^{\infty }\{{\displaystyle \sum _{s=0}^I{\mathcal{A}}_n{ℒ}_{qn}(x,y)}{e}^{i{\omega }_n^{+}t}+{\displaystyle \sum _{s=II}^{III}{ℬ}_n{ℒ}_{qn}(x,y)}{e}^{i{\omega }_n^{-}t}\}}$$

(32)

where the +ve sign corresponds to waves in FFM and the −ve sign corresponds to waves in the SFM. The waves in the FFM are generated due to the presence of the floating flexible plate, and the waves in SFM are generated by the presence of the submerged flexible porous plate. The values of ${\omega }^{+}$ and ${\omega }^{-}$ are the frequencies of the waves in the FFM and SFM, respectively. Further, ${ℒ}_{sn}(x,y)=\cos (n\pi x/l)f_{sn}({\alpha }_{sn},y)$, $f_{sn}({\alpha }_{sn},y), \mathrm{and} s=0,..,III$ are the same as in Equation (19), with ${\alpha }_{sn}={\{{(n\pi /l)}^2+{\gamma }^2\}}^{1/2}$, which satisfies the dispersion relation:

$$\tau =\frac{{\alpha }_{sn}({\kappa }_1{\alpha }_{sn}^4-{\epsilon }_1{\alpha }_{sn}^2+1)}{{\Lambda }_n},$$

(33)

where

$${\Lambda }_n=\frac{\tau (1+\tanh {\alpha }_{qn}d\tanh {\alpha }_{qn}H)-\vartheta ({\alpha }_{qn})\{i\sigma (1+\tanh {\alpha }_{qn}d\tanh {\alpha }_{qn}H)+\tanh {\alpha }_{qn}H\}}{\tau (\tanh {\alpha }_{qn}d+\tanh {\alpha }_{qn}H)-\vartheta ({\alpha }_{qn})\{i\sigma (\tanh {\alpha }_{qn}d+\tanh {\alpha }_{qn}H)+\tanh {\alpha }_{qn}d\tanh {\alpha }_{qn}H\}}$$

(34)

with $\vartheta ({\alpha }_{qn})={\alpha }_{qn}({\kappa }_2{\alpha }_{qn}^4-{\epsilon }_2{\alpha }_{qn}^2)$. The expression for ${\mathrm{\Lambda }}_n$ contains a ${\omega }^2$ term, thus solving Equation (32) for ${\omega }^2$ in terms of basin length and angle of incidence as ${\alpha }_{qn}={\{{(n\pi /l)}^2+{\gamma }^2\}}^{1/2}$. In the case of deep water, the dispersion relation (32) yields:

$${\omega }_n^{2+}=g{\alpha }_{qn}({\kappa }_1{\alpha }_{qn}^4-{\epsilon }_1{\alpha }_{qn}^2+1) \mathrm{and} {\omega }_n^{2-}=\frac{g{\alpha }_{qn}({\kappa }_2{\alpha }_{qn}^4-{\epsilon }_2{\alpha }_{qn}^2)(1+2i\sigma )}{2}$$

(35)

Where the + and − signs correspond to the waves in FFM and SFM, respectively. From Equation (34), the phase velocities, $v_n^{p+}$ in FFM and $v_n^{p-}$ in SFM, are found to be:

$$v_n^{p+}=\sqrt{\frac{g({\kappa }_1{\alpha }_{qn}^4-{\epsilon }_1{\alpha }_{qn}^2+1)}{{\alpha }_{qn}}} \mathrm{and} v_n^{p-}=\sqrt{\frac{g({\kappa }_2{\alpha }_{qn}^4-{\epsilon }_2{\alpha }_{qn}^2)(1+2i\sigma )}{2{\alpha }_{qn}}}$$

(36)

Further, the group velocities, $v_n^{g+}$ in FFM and $v_n^{g-}$ in SFM, are obtained as:

$$v_n^{g+}=\frac{(5{\kappa }_1{\alpha }_{qn}^4-3{\epsilon }_1{\alpha }_{qn}^2+1)\sqrt{g}}{2\sqrt{{\alpha }_{qn}({\kappa }_1{\alpha }_{qn}^4-{\epsilon }_1{\alpha }_{qn}^2+1)}} \mathrm{and} v_n^{g-}=\frac{(5{\kappa }_2{\alpha }_{qn}^4-3{\epsilon }_2{\alpha }_{qn}^2)\sqrt{g(1+2i\sigma )}}{2{\{2{\alpha }_{qn}({\kappa }_2{\alpha }_{qn}^4-{\epsilon }_2{\alpha }_{qn}^2)\}}^{1/2}}$$

(37)

Finally, the period of hydroelastic wave oscillations, $T_n^{+}$, of the floating flexible plate in FFM and $T_n^{-}$ of the submerged flexible porous plate in SFM are derived as:

$$\begin{gathered} T_n^{+}=\sqrt{\frac{4\pi }{g(n/a)({\kappa }_1{\alpha }_{qn}^4-{\epsilon }_1{\alpha }_{qn}^2+1)}}\mathrm{and} \\ T_n^{-}=\sqrt{\frac{8}{g(n/a{)}^2{\alpha }_{qn}({\kappa }_2{\alpha }_{qn}^2-{\epsilon }_2)(1+2i\sigma )}}. \end{gathered}$$

(38)

In the numerical results, the significant contributions to the effect of oblique incidence angle on the $v_n^{p+}$, $v_n^{p-}$, $v_n^{g+}$, $v_n^{g-}$, ${\omega }_n^{2+}$, ${\omega }_n^{2-}$, $T_n^{+}$, and $T_n^{-}$ in FFM and SFM are analysed by comparing their results propagating in FFM and SFM.

The numerical results are computed by considering water density=$\rho =1025 {\mathrm{kg}/\mathrm{m}}^3$, $d/h=0.25$, gravitational constant, $g=9.8{ \mathrm{ms}}^{-1}$, $E_1{I}_1=0.47\times {10}^3{\mathrm{Nm}}^2$, $E_2{I}_2=0.47\times {10}^2{\mathrm{Nm}}^2$, $N_1=1.2\sqrt{E_1{I}_1\rho g}$, $N_2=(1/2)\sqrt{E_2{I}_2\rho g}$, $\sigma =1.0+i$, and $n=1$ (n = 1 is considered for the sake of simplicity), if not otherwise specified).

Figure 13 compares the results of $v_n^{p+}$ and $v_n^{p-}$ for different values of oblique incidence angle, $\theta$, versus l(m) are plotted. The phase velocity, $v_n^{p-}$ , moves faster than those of $v_n^{p+}$ for a shorter basin with different values of $\theta$. However, in longer wave basins with lower values of $\theta$, the phase velocity, $v_n^{p-}$, moves slower than those of $v_n^{p+}$ due to the presence of compressive force and porosity in the submerged plate, which yields to the strong absorption of wave energy by the porous plate. On the other hand, at the edges of the basin, the variation of the effect of $\theta$ becomes significant in both the cases of $v_n^{p-}$ and $v_n^{p+}$.

Figure 14 compares the group velocity, $v_n^{g+}$ and $v_n^{g-}$, for different $\theta$ against l(m). Figure 14 reveals that $v_n^{g-}$ in SFM passes faster than those of $v_n^{g+}$ in FFM for almost all higher values of $\theta$. However, in a longer wave basin with lower values of $\theta$, the group velocity, $v_n^{g-}$, in SFM moves slower than that of $v_n^{g+}$ in FFM, for reasons similar to those in Figure 13. Further, it can be seen that the variations of $v_n^{g+}$ in FFM for different values of oblique angle, $\theta$, have no significant effect compared to $v_n^{g-}$ in SFM. This can be attributed to the fact that the wave energy is dissipated by the submerged flexible porous plate, while less incident wave transmits, which leads to more impact to $v_n^{g-}$ in SFM.

Figure 15 compares ${\omega }_n^{+}$ in FFM and ${\omega }_n^{-}$ in SFM for different angles, $\theta$, against l(m). It can be seen that, for higher values of the angle, $\theta$, both ${\omega }_n^{+}$ and ${\omega }_n^{-}$ increase. Further, the values of ${\omega }_n^{+}$ are lower than those in SFM, which is due to the presence of porosity in the submerged flexible plate that absorbs wave energy. Further, it is also observed that, for a longer basin, the effect of oblique angle, $\theta$, on the wave dispersion is significant while, for a shorter basin, the effect of $\theta$ is negligible.

In Figure 16, the period of oscillations, $T_n^{+}$ and $T_n^{-}$, decreases with an increase in the values of $\theta$. This is due to the fact that, as the incident angle changes, the direction and types of the oscillation also change with their mode shapes, which results in a lower period of oscillation. However, the effect of $\theta$ is significant for a longer basin, and the vibration of the floating plate becomes less, possibly due to the wave energy absorbed by the porous plate.

Figure 17 shows the effect of the period of oscillation, $T_n^{-}$, on the increasing values of the porous-effect parameter versus basin length, $l(m)$, for different angles of incidence,$\theta$. It can be seen that the period of oscillations, $T_n^{-}$, decreases with increasing values of $\theta$. In the case of $\theta =0^0$, the observation is similar to that obtained in [34]. However, the period of oscillation experienced peaks for zero incidence angle at around 10–15 m, which may be due to phase change of incident and reflected waves, leading to constructive/destructive interference of the waves in SFM. The decreasing rate of oscillation is much faster for increasing $\theta$, and as the basin length increases, the period of oscillation reaches a smooth trend. One of the reasons for this may be the increasing rate of energy dissipation along the porous plate.

Figure 18 shows the effect of the first six modes, $n$, in the horizontal direction, on the period of oscillation, $T_n^{+}$, of the floating flexible plate with incidence angle $\theta ={30}^0$ versus basin length $l(\mathrm{m})$. It is observed that the variations of the period of oscillations continue to decrease with an increase in basin length. This indicates that the modes of oscillation have an effect at the edges of the plate for a certain value of the incident angle, which changes the shape of the floating plate.

5. Conclusions

In this paper, a new contribution complementing [34] addresses the influence of oblique incidence angle based on the reduced wave equation. The hydroelastic analysis of a moored floating and a submerged flexible porous plate is investigated by studying the effect of the incident angle for different design parameters. Further, the free oscillation of hydroelastic waves in a wave basin is performed in specific cases. From the analysis, it has been observed that:

• The convergence of the obtained solution is verified for different oblique incidence angles, and $N=6$ is confined for computational accuracy.
• It is found that the obtained results are supported by the published experimental and numerical FEM-BEM results.
• For small oblique incidence angles (θ < 55°), $C_r$ increase for higher values of mooring stiffness and the compressive force of the floating plate with all parameters except porous-effect parameter associated with submerged porous plate, while $C_t$ and $C_e$ follow the reverse pattern. However, for θ = 90°, the incident wave is normal to the floating and submerged flexible porous plates; it travels along with the plate in the z-direction, while the reflection $C_r$ decreases and $C_e$ increases for higher values of porosity.
• The displacement results showed that, for compressive force and submergence depth, the amplitude of the plate displacement at the edges continues increasing as the incident angle increases, occurring a higher number of crests on the plate surface. On the other hand, it is observed that the submerged porous plate deformation decreases with an increase in mooring stiffness.
• In a closed wave basin, for the higher angle of incidence, the period of oscillations of hydroelastic waves over FFM and SFM decreases, while the ${\omega }_n^{\pm }$, $v_n^{p\pm }$, and $v_n^{g\pm }$ increase. Further, $v_n^{p-}$ and $v_n^{g-}$ move faster than that of $v_n^{p+}$ and $v_n^{g+}$, having a significant effect of oblique incidence angle in a longer wave basin. The analysis of the results will be useful in the laboratory testing and design of a wave tank.
• The present study indicated that the appropriate choice of oblique incidence angle, mooring stiffness, compressive force, and submergence depth will be useful to design a flexible plate system for an effective breakwater and the reduction of the hydroelastic response of floating flexible structures.
• The limitations of the present study are that the relevant boundary conditions must be linear and of higher-order, maybe 3rd or 5th order, and the structural boundaries must be a constant of the co-ordinates; hence, the solution must be a series of solutions. Therefore, the present approach can be used for rectangular and circular-shaped flexible/porous plate and membrane type structures.
• Further, the model formulation and calculation allow analysis of the hydrodynamic loads (horizontal and vertical) on the structure and the axial forces on the mooring lines.