A mathematical study of water wave interaction with a thin perforated barrier in a two-layer fluid over a permeable bottom

Abstract

This article studies the impact of a submerged interface-piercing perforated barrier in a two-layer fluid flowing over a permeable bottom. We investigate oblique wave scattering, trapping and radiation due to the structure focusing on the bottom permeability. The dead water phenomenon is analysed with the consideration of the bottom permeability, which results in a higher variation of the interfacial wave due to the bottom permeability. The matched eigenfunction expansion method and the least square technique are used to calculate various hydrodynamic coefficients. Wave energy identity relation is derived for the scattering scenario, and the associated energy loss due to the barrier is calculated. In order to attain the maximum wave dissipation, an ideal porous-effect parameter of the barrier is proposed for consideration, and it is observed that larger values of porous-effect parameter result in the lowest feasible pressure distribution. A good comparison with a prior result justifies the current semi-analytical procedure. Furthermore, the verification of the energy-identity terms aid in the validation of the computed results. Additionally, wave trapping in a confined region is examined by investigating reflection coefficients by considering a rigid wall. The thin perforated barrier model is further considered for examining the radiation aspect while considering its slow motion. For various porous-effect parameters of the barrier, the amplitude ratio of the radiated potential is investigated, and it is clearly observed that higher frequency significantly lowers the amplitude for both free surface and interfacial propagating modes. The impact of the perforated barrier is analysed by investigating the essential hydrodynamic coefficients, namely, added mass and damping coefficient.

Background and introduction

Important works and motivation

The interaction of water waves with a submerged or floating impediment, such as a breakwater, has a variety of marine engineering applications. There has been an increasing interest in investigating this phenomenon since the 1960s. Porous media are increasingly being used in science and engineering. Filtration, acoustics, geo-mechanics, soil mechanics, rock mechanics, petroleum engineering, hydro-geology, geophysics and biophysics are just a few disciplines that exhibit numerous benefits of use of porous media. Fluid transfer, particularly across porous media, has piqued the curiosity of many scientists and engineers. Perforated structures reduce the impact of waves on harbours, ports, inlets and other structures, rendering the wave behaviour much calmer. Perforated structures are hence used as breakwaters to protect harbours, ports, inlets and beaches from extreme wave impact in coastal areas, including shoreline and harbour regions. These assure the safety and stability of various activities, including cargo handling at ports.

Sollitt and Cross ¹ came up with the first known wave-induced porous medium flow model in 1972. Later, the potential theory was employed to solve boundary value problems for the scattering of waves in a porous medium.^2,3 For rubble-mound breakwaters, Sulisz ⁴ used the boundary element method to build an approximation of the correct solution. Dalrymple et al. ⁵ were the first ones to observe that the dispersive roots in a porous medium might overlap with one another, and they devised the Green’s function approach to deal with this problem. Chwang ⁶ presented a pioneering work in developing a thin porous breakwater model based on thin piston-type porous wavemaker theory. The thin porous breakwater model was developed and used by Yu, ⁷ Lee and Chwang, ⁸ Sahoo et al.,^9,10 Manam and Sahoo, ¹¹ Behera et al., ¹² Panduranga et al. ¹³ and Zhao et al. ¹⁴ etc. Barman and Bora ¹⁵ examined the impact of a flexible structure as a breakwater in finite ocean depth for a two-layer fluid. However, the impact of diffraction and radiation in the presence of thin perforated structures is not available in the literature as per the best knowledge of the authors.

A more realistic situation of water waves interacting with perforated materials in a stratified ocean has not been appropriately addressed earlier, which now needs to be highlighted. The density ratio for a two-layer fluid containing freshwater and salty water should ideally be greater than $0.97$ . Significant fluctuations in ocean water temperature and salinity cause a significant shift in ocean water density, resulting in the formation of these numerous strata or layers. To put it another way, ocean water comprises various strata with different densities and abrasive surfaces. The interfaces between distinct strata become unstable as the density gradient fluctuates. Furthermore, wave-like disturbances called interfacial waves are created when an impulse is generated. The monograph by Massel ¹⁶ explains the importance of interfacial waves in detail. In addition, the restoring force for water waves – relative buoyancy force – is proportional to the product of gravity and the density difference between the two strata. At the interface between two layers, the difference in density between the layers is much smaller than the difference in density between air and water. Consequently, internal waves generated at the interface may have much larger amplitudes than free surface waves. Internal waves are regarded to be a cause of concern to ships and other offshore structures due to their large magnitudes. Russell^17,18 was the first to notice such a finding. Osborne et al. ¹⁹ observed the influence of internal waves and deep-sea drilling. More research in different problems in two-layer fluids can be found in Behera et al., ¹² Behera and Sahoo ²⁰ and Barman and Bora.^21,22 There are numerous examples of two-layer fluid systems, including glacier lubrication by a subglacial hydrological system Fowler, ²³ the flow of stratified layers of glacial material with differing properties Loewenherz et al., ²⁴ deformation of till beneath glaciers (e.g. Clarke ²⁵ and MacAyeal ²⁶ ).

This theoretical investigation attempts to look at the effects of a vertically placed perforated barrier on water waves propagating in a two-layer fluid across a permeable bottom. The wave-structure interaction problem is solved by using the matched-eigenfunction expansion and the least square approach. The dead water analogue is addressed in the light of the bottom permeability consequences. The waveload, reflection coefficients and other physical variables are then computed. For such a complex model, the energy identity is derived with the help of Green’s integral approach, and the role of the barrier in energy dissipation is conscientiously examined. By considering different physical values, the angle of incidence and other physical factors related to the geometric configuration, we investigate the hydrodynamic coefficients. A rigid wall is considered at a finite distance from the perforated barrier, which sheds light on wave trapping in the confined region. Additionally, the radiation aspect is investigated and assessed to concentrate on a few crucial components of the barrier problem while considering a small amplitude motion. To the best of the authors’ knowledge, a permeable sea-bed has not been taken into account in any work with respect to the interaction of gravity waves with perforated substances in a two-layer fluid. The results of this research are expected to influence how marine facilities are to be designed in order to lessen the wave power that strikes the infrastructure.

Practical applications

The ‘thin’ perspective issue appears here due to the consideration of comparative length scale in the domain. The whole geometry in considered in large scale $(-\infty ,\infty )$ or $(-\infty ,L)$ . In comparison to the large length scale of the whole domain, the thickness of the breakwater is very small. The ‘thin’ assumption is one of the perspectives based on this consideration. The theory of thin structures was developed by Chwang ⁶ . Moreover, the application of such a structure as a protector of waves in coastal or reservoir density current is a perfect practical example. However, it may be noted there are many other examples as well. Perforated breakwaters have become increasingly popular as anti-reflective quay walls or external dykes. Most are composed of a vertical perforated (or slotted) screen. Perforated vertical breakwaters are intended to absorb part of the wave energy through various mechanisms, such as turbulence, viscous friction and wave resonance. Moreover, the drawbacks of massive rubble mound breakwaters and caissons include high prices, significant benthic footprints, adverse environmental effects and inadequate deep water depth. As a result, scientists have investigated using thin, porous plates as a potential perforated breakwater idea. Both horizontal and vertical plates may function as breakwaters by reflecting waves and dissipating their energy as they travel through the pores of the plates. Additional wave energy loss occurs when the plates are submerged because of the shear friction caused when water moves between the plate and the free surface (see Neelamani and Reddy ²⁷ ). The plate-type perforated barrier may be erected as a stand-alone breakwater or as an add-on structure to the existing offshore construction. The benefit of a single-layer porous plate breakwater is its affordable construction. Compared to solid plates, the porous design has effectively reduced the vertical wave force. ²⁸ The porous horizontal scale is a practical choice for a breakwater since theoretical investigations have shown that an ideal wave attenuation performance may be obtained with a plate width-to-wavelength ratio of around $0.25$ .^29,30 When geometrical parameters were tuned, Liu et al. ²⁹ found that a wave transmission coefficient of $0.55$ and below could be attained on two-layer horizontal porous plates. Theoretical investigation on vertical barriers can be observed in the works of Manam and Sivanesan, ³¹ and Mackay and Johanning. ³² Huang et al. ³³ showed that multi-layered vertical porous plates may achieve a wave transmission coefficient having values as small as $0.2$ in terms of actual performance while also having a very low wave reflection coefficient.

More complex designs, including perforated plates, have also surfaced to enhance the performance of breakwaters. The Jarlan-type caisson, which consists of layers of vertical porous plates erected in front of a perforated caisson and is often filled with porous materials like sand, is a well-known design example. The caisson and plates will trap the input wave, increasing the local wave amplitude and resulting in wave energy dissipation. Wang et al. ³⁴ illustrated a Jarlan-type caisson in their research. Neelamani et al. ³⁵ examined a double-layered slotted wall prototype for a perforated barrier. Moreover, surface waves in reservoirs or lakes induced by landslides during earthquakes are being studied by using the thin porous-wavemaker hypothesis, which has numerous applications in the research of surface waves in such water bodies. Landslides produce water waves by moving vertically at an angle or horizontally during the part of their journey. A vertical landslide was simulated by Noda ³⁶ by using a two-dimensional box lying at the bottom at the end of a semi-infinite canal. Consideration of a porous box or wall would be more suitable when the majority mass of a landslide comprises rocks and soils. Another possible use of the current approach is when the barrier is subjected to some structural limitation on the highest permitted force, and the efficiency of the barrier is of primary importance. As the porous-effect parameter grows, a perforated barrier may help in lowering the overall load and dropping the wave amplitude. The thin plate inhibits the vertical motion of water oscillations so efficiently that it may significantly lower wave amplitudes. Wu et al. ³⁷ proposed an engineering application in which a horizontal plate might be connected to an existing sea-wall to lessen wave reflections. A pitching plate may be designed as part of a wave controller, according to Yip and Chwang. ³⁸ They further considered a porous wall breakwater with an internal horizontal plate. Apart from the one-layer fluid flow problem, there are many instances of two-layer fluid flow systems with thin structure interaction problem (See Refs.^{11,12,39–41}) All these works consist of thin porous structures interacting with two layer fluid using linear water wave theory.

Another fundamental problem is that numerous porous breakwaters have the requirement to buoy themselves. These breakwaters must be supported by foundations, often permanent and difficult to remove. Hybrid designs of permeable and floating breakwaters have been developed to address this drawback. This allows for the use of a lighter mooring system. The floating breakwater ideas are divided into seven forms in the review article by Dai et al. ⁴² : box, pontoon, frame, mat, tethered float, horizontal plate and others. When paired with the porous breakwater ideas, these floating breakwaters combined the benefits of both kinds to provide several innovative designs that had both mobility and improved energy dissipation. For instance, Qiao et al. ⁴³ affixed slotted walls to a floating breakwater as a box. Under a floating dual-pontoon breakwater, Ji et al. ⁴⁴ erected nets and filled a porous substance. In order to provide a barrier against the waves, Kou et al. ⁴⁵ connected an outer layer of ring-shaped slotted walls to the floating VLFS. Wang and Sun created a unique porous breakwater made of porous diamond-shaped blocks in their 2010 experiment (See Wang and Sun ⁴⁶ ). Due to their modest weight and ability to float, these blocks may be utilised to maintain stations using mooring lines rather than a gravity-based foundation. A unique floating breakwater was presented by Wang et al. ⁴⁷ with layers of L-shaped tubes running through the breakwater hull, enabling water to oscillate within the tubes and further dissipating wave energy. Floating breakwaters have shown improved hydrodynamic capabilities using porous components, including decreased wave transmission, lower wave reflection and decreased wave stresses on the structure. In this article, we model a thin perforated barrier along with the consideration of the bottom permeability. Looking at the applications as detailed above, the current work is expected to aid in additional engineering applications.

Mathematical formulation

The interaction of surface waves with a vertically placed floating perforated barrier in a two-layer fluid over a permeable sea-bed is investigated in this paper by employing linear water wave theory. In Figure 1(a), the geometric sketch of the model is set up in a right-handed Cartesian system with the $z$ -axis pointing upward such that the free surface and the bottom surface are indicated by $z=0$ and $z=-H$ , respectively. An incompressible fluid of two immiscible layers is considered with inviscid and irrotational motion. The interface between the fluid layers is assumed at $z=-h$ . The barrier is considered at $x=0$ in $-a\le z\le 0$ , dividing the whole domain into Regions 1 and 2. With gravity in consideration, the obliquely incident surface wave impinges on the barrier at an angle $\theta$ measured from the $x$ -axis. Considering time-harmonic motion, the velocity potentials are represented as ${\Phi }_m(x,y,z,t)=\mathrm{Re}[{\varphi }_m(x,z)e^{\mathrm{i}(k_{y}y-\omega t)}]$ , $m=1,2$ where $m=1$ refers to Region 1 and $m=2$ refers to Region 2. Here, $k_y=k_1\sin \theta$ is the $y$ -component of the progressive wave mode $k_1$ propagating along the free surface, ‘Re’ stands for the real component of the quantity $\left[\right]$ and $\mathrm{i}$ stands for the imaginary quantity. Following Snell’s Law, it is considered that the $y$ -directional velocity potentials are equal along the vertical boundaries. The densities of the fluid in Region 1 (upper) and Region 2 (lower) are considered to be ${\rho }_1$ and ${\rho }_2$ , respectively.

Figure 1.

Sketch of (a) wave scattering model with a perforated barrier in geometric cross-section, (b) three-dimensional view and (c) wave trapping model in geometric cross-section.

For layer-wise elucidation, ${\varphi }_m$ are spatially dispersed along the $z$ -direction as

$${\varphi }_m=(\begin{array}{cc}{\psi }_m^u,\hfill & z\in (-h,0],\hfill \\ \multicolumn{1}{c}{{\psi }_m^l,}& z\in [-H,-h),\hfill \end{array}\mathrm{for}m=1,2,$$

(1)

where the potentials corresponding to the upper and lower layers are denoted by the superscripts $u$ and $l$ , respectively.

For forming and solving the boundary value problems (BVP), the modified Helmholtz equation is to be used as the governing equation:

$$\begin{array}{cc}({\nabla }^2-k_y^2){\psi }_m^u=0,\hfill & z\in (-h,0),\hfill \\ \multicolumn{1}{c}{({\nabla }^2-k_y^2){\psi }_m^l=0,}& z\in (-H,-h),\hfill \end{array}\}\mathrm{for}m=1,2,$$

(2)

where ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial z^2}$ .

At $z=0$ , the free surface condition yields

$$\frac{\partial {\psi }_m^u}{\partial z}-K{\psi }_m^u=0,m=1,2,$$

(3)

where $K={\omega }^2/g$ with $g$ as the acceleration due to gravity.

We obtain the boundary condition at the bottom $z=-H$ as follows:

$$\frac{\partial {\psi }_m^l}{\partial z}+\mathcal{G}{\psi }_m^l=0,\mathrm{at}z=-H,\mathrm{for}m=1,2,$$

(4)

where $\mathcal{G}$ is the permeability of the bottom (See Refs.^48–50).

Since the lower layer fluid has a higher density and hence considering $\rho (={\rho }_1/{\rho }_2),0<\rho <1$ , the boundary conditions at $z=-h$ are as follows:

$$\begin{array}{c}\frac{\partial {\psi }_m^u}{\partial z}=\frac{\partial {\psi }_m^l}{\partial z},\hfill \\ \multicolumn{1}{c}{\rho (\frac{\partial {\psi }_m^u}{\partial z}-K{\psi }_m^u)=(\frac{\partial {\psi }_m^l}{\partial z}-K{\psi }_m^l),}\end{array}\mathrm{for}m=1,2.$$

(5)

Perforated barrier condition

There are various theories already available with respect to the perforated barrier. If we consider the perforated barrier as a thick porous plate consisting of small and uniform pores, then following Yu, ⁷ the thick plate assumption can be converted to a thin one by doing non-dimensional analysis along the $x$ -dimensional propagation. Then the matching conditions over the perforated barrier yield

$$\begin{array}{c}\frac{\partial {\psi }_1^u}{\partial x}=\frac{\partial {\psi }_2^u}{\partial x}=\mathrm{i}{k}_1{\mathcal{G}}_1({\psi }_1^u-{\psi }_2^u),z\in (-h,0)\hfill \\ \multicolumn{1}{c}{\frac{\partial {\psi }_1^l}{\partial x}=\frac{\partial {\psi }_2^l}{\partial x}=\mathrm{i}{k}_1{\mathcal{G}}_1({\psi }_1^l-{\psi }_2^l),z\in (-a,-h)}\end{array}\}\mathrm{at}x=0,$$

(6)

where ${\mathcal{G}}_1=\frac{ϵ}{k_{1}b\gamma }$ . The parameter $\gamma =f-\mathrm{i}S$ is the porous impedance parameter where $S=1+\frac{1-ϵ}{ϵ}{C}_m$ with $ϵ$ as porosity, $C_m$ as added mass coefficient and $f$ as the friction parameter. Such assumption has already been made and successfully employed by Isaacson et al., ⁵¹ Sahoo et al.,^9,10 Yip and Chwang, ⁵² Li et al.,^53–55 and Williams et al. ⁵⁶

Furthermore, the potentials satisfy the continuity of velocity and pressure along the gap at $x=0$ as

$$\frac{\partial {\psi }_1^l}{\partial x}=\frac{\partial {\psi }_2^l}{\partial x},{\psi }_1^l={\psi }_2^l,\mathrm{at}x=0,z\in (-H,-a).$$

(7)

Rigid wall condition

When waves encounter coastal structures or sea walls, they lose substantial energy via run-up and reflection. Additionally, the amplitude of the incident wave gets boosted near the sea wall by a maximum amount of twice the height of the wave. Because of the scouring that occurs at the base of the wall, safety measures must be contemplated to prevent the wall from collapsing. Therefore, efforts are being made to provide a quiet zone with reduced stress on the existing infrastructure by using perforated structures at a certain distance from the sea wall or coastal infrastructures, thus reducing the potency of the surface gravity waves. The wave trapping model considered next, as shown in Figure 1(c), is thus also taken into consideration in order to investigate the bottom and barrier effect and reduce the force on the wall. The potential ${\varphi }_2$ additionally satisfies the no-flow condition at the rigid wall $x=L$ as follows:

$$\begin{array}{c}\frac{\partial {\psi }_2^u}{\partial x}=0,\mathrm{for}z\in (-h,0),\hfill \\ \multicolumn{1}{c}{\frac{\partial {\psi }_2^l}{\partial x}=0,\mathrm{for}z\in (-H,-h).}\end{array}$$

(8)

Plane wave approximation

Progressive wave solution

Following the governing equation (2) and the respective boundary conditions (3)−(5), the velocity potentials ${\varphi }_m$ after combining ${\psi }_m^u$ and ${\psi }_m^l$ can be expressed as follows:

$${\varphi }_m=\exp (\pm \mathrm{i}\mathrm{xq})\mathcal{Z}(k,z),z\in (-H,0),$$

(9)

with $q=\sqrt{k^2-k_y^2}$ , and eigenfunctions $\mathcal{Z}(k,z)$ given by

$$\mathcal{Z}=(\begin{array}{cc}\frac{\mathcal{P}(k)(K\sinh \mathrm{kz}+k\cosh \mathrm{kz})}{(K\cosh \mathrm{kh}-k\sinh \mathrm{kh})},\hfill & z\in (-h,0),\hfill \\ \multicolumn{1}{c}{\cosh k(z+H)-\frac{\mathcal{G}}{k}\sinh k(z+H),}& z\in (-H,-h),\hfill \end{array}$$

(10)

where $\mathcal{P}(k)=(\sinh k{H}_1-\frac{\mathcal{G}}{k}\cosh k{H}_1)$ with $H_1=(H-h)$ .

Following the second equation of (5), we obtain the dispersion relation

$$\begin{array}{c}\mathcal{D}(k)\equiv [K^2(\rho +\coth \mathrm{kh}\coth k{H}_1)-\mathrm{kK}(\coth \mathrm{kh}+\coth k{H}_1)\hfill \\ +k^2(1-\rho )]-\mathcal{G}[(1-\rho )k^2\coth k{H}_1+K^2(\rho \coth k{H}_1\hfill \\ +\coth \mathrm{kh})-\mathrm{kK}(\coth \mathrm{kh}\coth k{H}_1+1)]=0,\hfill \end{array}$$

(11)

where $k$ is the dispersive root or propagating mode.

Discussion of the dispersion relation

The dispersion relation (11) yields two pairs of real symmetric roots, $k=\pm k_1,\pm k_2$ , constrained by $0<k_1<k_2$ . It validates the presence of two progressive wave modes: (i) Free Surface Mode (SM) and (ii) Interfacial Mode (IM). Moreover, this equation contains an infinite number of conjugate pairs of symmetric purely imaginary roots such as $k=\pm k_i=\pm \mathrm{i}{t}_i$ for $i=3,4,$ , with $t_i$ as a real integer, which correspond to evanescent modes (See Behera et al. ⁵⁷ ). Furthermore, the negative values of those real and purely imaginary roots represent wavenumbers of waves travelling in the negative direction. To find the roots of the dispersion relation (11) is a fairly typical job, and hence it is not covered in this work. The dispersion relation for a homogeneous fluid system can be found by considering $\rho =1$ and $h=H$ . When the permeability of the bottom vanishes, that is, $\mathcal{G}=0$ , 2.3 results in the dispersion relation for a rigid bottom.

Finding the roots of the dispersion relation with complex $\mathcal{G}$ , on the other hand, is more difficult. The dispersion relation for complex $\mathcal{G}$ has complex roots given by $k_i=\pm (a_i+\mathrm{i}{b}_i)$ for $i=1,2$ , where all $a_i$ and $b_i$ are real. The roots nearer the real axis correspond to the most prominent progressive waves in surface and interfacial modes. It also contains an infinite number of complex roots near the imaginary axis as $\pm k_i,i=3,4$ , which correspond to the evanescent modes. In order to identify the roots, the algorithm of Mendez and Losada ⁵⁸ is followed so that the roots get transformed from real $\mathcal{G}$ to complex $\mathcal{G}$ . The benefit of this methodology is that the propagating modes can be differentiated, even though they are all complex. Other techniques, such as the homotopy perturbation (HP) method of Chang and Liou, ⁵⁹ can also be applied. Barring the number of iterations and the specified error in $\mathrm{Kh}$ , Barman and Bora ²¹ found that the roots derived by the two approaches as mentioned above do not differ significantly from each other up to the practical precision level.

Furthermore, it was found that the HP technique attained convergence faster, corresponding to the same number of iterations and with a tiny inaccuracy in terms of the values of the wavenumber. The methodology of Mendez and Losada ⁵⁸ can be adopted due to the consistency of the results in both ways, ignoring the computational cost. Furthermore, the direct application of the Newton-Raphson-based algorithm through Matlab would provide with the roots. Still, it may be accompanied by a reasonable amount of complication while guessing the suitable initial value. In addition, for $\mathcal{D}\prime (k)=0$ , with ′denoting differentiation, the Newton-Raphson method does not succeed. Similarly, the eigenfunction expansion also fails for these values of $k$ . In this context, we apply a programme-based function in Matlab to verify the complex roots. For this article, we obtain the real and purely imaginary roots for real $\mathcal{G}$ by a developed Matlab code based on the method of Regula Falsi.

Effect of bottom permeability on dead water analogue

One of the fascinating features of two-layer fluids is the dead water phenomenon, which creates excessive wave resistance in ships. Since minimal changes in potential energy are needed to produce internal waves, an imposed disturbance, which slowly moves on or below the free surface so that its free-surface signature can hardly be noticed, may be very effective in generating large-amplitude waves on the interface. This effect is closely related to the so-called ‘dead-water’ phenomenon, where slow ships often lose their speed and steering capabilities when moving in regions of strong density stratification. The density stratification may occur due to salinity or temperature. Physically it is an analogue which results in a high-amplitude internal wave with a low amplitude at the surface. This usually occurs irrespective of water depth when the densities of the two fluids are close to each other. This phenomenon is well-known to sailors. This type of still water situation where it is difficult to row a boat was mentioned in the Latin literature by Tacitus, who experienced such problems in Scotland and Germany (see Mercier et al. ⁶⁰ ). However, the Norwegian Arctic explorer Fridtjof Nansen provided the first well-documented report on the dead water phenomenon (see Nansen ⁶¹ ), who encountered this phenomenon while sailing in his ship Fram near Nordenskild island in 1893. A proper scientific explanation of such abnormal behaviour was not provided until Ekman, ⁶² motivated by the report of Nansen, investigated the phenomenon in detail with the help of experiments. His experiments confirmed that the massive resistance experienced by the slowly moving ship was mainly because of the generation of internal waves and that the maximum drag resistance occurred when the speed of the ship matched with that of the internal waves. Miloh et al. ⁶³ developed a linear explanation for the occurrence of dead water in a two-layer fluid with restricted depth. A nonlinear theory was developed by Grue et al. ⁶⁴ Due to the dead water analogue, two drag phenomena were observed by scientists. The first, Nansen ⁶¹ wave-making drag, causes a constant, abnormally low speed. The second, Ekman ⁶² wave-making drag, is characterised by speed oscillations in the trapped boat. Recent research shows that these speed variations are due to the generation of specific waves that act as an undulating conveyor belt on which the ship moves back and forth. The scientists have also reconciled the observations of both Nansen and Ekman. They have shown that the Ekman oscillating regime is only temporary: the ship escapes and reaches the constant Nansen speed. Moreover, when the densities of both layers are very close, the energy concentration at the interface will be extremely high compared to that on the free surface. A significant area of the theory is to examine the impact on floating structures such as VLFS (very large floating structures) or breakwaters, which are large thin structures designed to utilise ocean space for various purposes. Especially in tropical regions where salinity and temperature differences stratify the water Yuan et al., ⁶⁵ this theory would apply to mathematically model wave-structure interaction systems.

Other dead-water related works

Hughes and Grant ⁶⁶ took advantage of the dead-water effect to study the impact of interfacial waves on wind-induced surface waves. The study relates the statistical properties of surface waves to the currents induced by internal waves. Maas and van Haren ⁶⁷ investigated if the dead-water effect could also be experienced by swimmers in a thermally stratified pool, offering a plausible explanation for unexplained drownings of experienced swimmers in lakes during the summer season, but found no effect. One can argue that the stratification considered might have been inadequate for swimmers to generate waves and mainly led to the thermocline mixing. An energetic budget is given in a more detailed and idealised study Ganzevles et al., ⁶⁸ where the authors also observed some retarding effects on the swimmers. In a slightly different perspective, Nicolaou et al. ⁶⁹ demonstrated that an object accelerating in a stratified fluid generates oblique and transverse internal waves; the latter can be decomposed as a sum of baroclinic modes with the lowest mode always present. Shishkina ⁷⁰ further showed through experiments that the baroclinic modes generated propagate independently in such a dynamical evolution. However, bottom permeability effects become necessarily significant when the amplitude of the internal waves increases.

We try to illustrate the dead water analogue as well by examining the impact of bottom permeability in it. We consider one-dimensional wave form of the amplitudes ${\chi }_n={\eta }_n{e}^{-\mathrm{i}\omega t}$ for $n=1,2$ along the free surface and interface, respectively. Now, the velocity potentials can be expressed in terms of ${\eta }_i$ as

$$\Phi =(\begin{array}{cc}\frac{\mathrm{i}\omega }{k}(\Lambda \cosh \mathrm{kz}-\sinh \mathrm{kz}){\chi }_1,\hfill & \mathrm{for}z\in (-h,0),\hfill \\ \multicolumn{1}{c}{\frac{\mathrm{i}\omega (k\cosh k(z+H)-\mathcal{G}\sinh k(z+H))}{k(\mathcal{G}\cosh k{H}_1-k\sinh k{H}_1)}{\chi }_2,}& \mathrm{for}z\in (-H,-h),\hfill \end{array}$$

(12)

where

$$\Lambda =[\frac{\rho (1-\stackrel{2}{\coth }\mathrm{kh})}{(\frac{\mathrm{gk}(1-\rho )}{{\omega }^2}-\rho \coth \mathrm{kh})-\frac{(k\coth k{H}_1-\mathcal{G})}{(k-\mathcal{G}\coth k{H}_1)}}-\coth \mathrm{kh}].$$

Furthermore, the amplitude ratio is expressed as

$$\vartheta =\frac{{\eta }_2}{{\eta }_1}=\frac{\rho }{\mathcal{E}},$$

(13)

where $\mathcal{E}=((\rho \cosh \mathrm{kh}-\frac{\mathrm{gk}(1-\rho )}{{\omega }^2}\sinh \mathrm{kh})+\frac{(k\coth k{H}_1-\mathcal{G})}{(k-\mathcal{G}\coth k{H}_1)}\sinh \mathrm{kh})$ .

Equation (11) can be represented as a quadratic equation in ${\omega }^2$ as follows:

$$\mathrm{a}{\omega }^4-\mathrm{b}{\omega }^2+\mathrm{c}=0,$$

(14)

where

$$\begin{array}{c}\mathrm{a}=k(\rho +\coth \mathrm{kh}\coth k{H}_1)-\mathcal{G}(\rho \coth k{H}_1+\coth \mathrm{kh}),\hfill \\ \mathrm{b}=\mathrm{gk}[k(\coth \mathrm{kh}+\coth k{H}_1)-\mathcal{G}(\coth \mathrm{kh}\coth k{H}_1+1)],\hfill \\ \mathrm{c}=g^2{k}^2(1-\rho )(k-\mathcal{G}\coth k{H}_1).\hfill \end{array}$$

Equation (14) results in two propagating modes, such as

$$\begin{array}{c}{\omega }_1=\pm \sqrt{\frac{\mathrm{b}+\sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}},{\omega }_2=\pm \sqrt{\frac{\mathrm{b}-\sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}},\hfill \end{array}$$

(15)

where ${\omega }_1$ and ${\omega }_2$ signify the free surface and interface mode, respectively. Now, $\underset{\rho \to 1}{lim}{\omega }_2\to 0$ and $\underset{\rho \to 1}{lim}\vartheta \to \infty \mathrm{for}\omega ={\omega }_2.$ It clearly shows that the density stratification leads to the rise of the interfacial wave.

Figure 2(a) and (b) illustrate the impact of bottom permeability and wavelength in dead water analogue. The amplitude ratio is quite large as $\rho \to 1$ , which can be explained by the effect of dead water. While increasing the bottom permeability, we can observe an undulating pattern. Initially $\log \vartheta$ smoothly increases and after obtaining the value around $\mathcal{G}H>0.5$ , it increases sharply. However, after certain value of $\mathcal{G}H$ , no variation can be observed in the $\log \vartheta$ . Irrespective of the wave interaction system, it shows that the dead-water effect is also affected by a bottom permeability. After that, a steady behaviour can be observed. The undulating pattern due to bottom permeability and then the steady pattern also point towards the fact that there may be effect of bottom permeability on Ekman dragg effect. ⁶² Moreover, the steady nature of $\log \vartheta$ may be due to the fact that the wave dissipation due to bottom permeability exists upto some level. Once it attains the optimum value, steady nature of dissipation can be ascertained. However, higher bottom permeability results in comparatively smaller impacts in dead water analogue. The resistance effect of the bottom ensures a higher amount of wave reflection, which increases the wave energy along the interface. Therefore, the permeability of the bottom increases the density stratification impact on waves.

Figure 2.

Non-dimensional ratio of amplitude $\vartheta$ against $\rho$ corresponding to (a) non-dimensional permeability $\mathcal{G}H$ with $\mathrm{kH}=0.5$ , (b) non-dimensional $\mathrm{kH}$ with $\mathcal{G}H=5$ and $h/H=0.5$ and corresponding to non-dimensional $h/H$ for (c) $\mathcal{G}H=2.5$ and (d) $\mathcal{G}H=5$ with $\mathrm{kH}=0.25.$

Furthermore, while varying the wavenumber, we can observe that the dead water impact decreases for the case of deep water waves as compared to shallow water waves. This is due to the lower impact of $\mathcal{G}H$ and the higher flow along the free surface, for which a significant energy concentration can be observed near the free surface. However, for shallow water waves, the energy contribution is uniform and the bottom permeability dissipates the free surface energy.

In Figure 2(c) and (d), the elevation ratio is illustrated for two different values of $\mathcal{G}H$ . While varying the interface position, we can observe that, when the interface is closer to the bottom, the effect of dead water decreases, whereas it is the highest when the interface is closer to the free surface. We can assume that the free surface and interfacial interaction play a significant role in such amplitude hike. For $\mathcal{G}H=2.5$ , we can observe a smooth decay while varying interfacial position. However, an undulating decreasing pattern can be observed for $\mathcal{G}H=5$ . While the interface is closer to the bottom, the bottom permeability dissipates a significant portion of the interfacial energy. This may result in a sharp decrease in the elevation. However, while increasing density, the waves due to the surface-wave mode are negligible, but large waves are generated on the interface through the internal-wave mode. The conditions in these plots correspond to the well-known dead-water phenomenon where a significant wave drag can be experienced due to the generation of large amplitude internal waves.

Further, all the graphs depict that, when the densities of both layers are very close, the energy concentration at the interface will be extremely high compared to that on the free surface. Nevertheless, the dead-water phenomenon does not correspond to a constant amplitude ratio evolution. The dynamical study of the problem is thus much more prosperous than in the rigid bottom case. However, it is essential to mention that the bottom permeability implies changing the effective force on the wave interaction system. Without permeability, the effect of the bottom would be a constant one. The permeability is not included in previous studies and seems to be an essential characteristic of the dead-water phenomenon.

Scattering problem

Method of solution

Our main aim is to utilise the eigenfunction expansion method to find solutions by using the orthogonality of eigenfunctions. In the scattering problem, we can express the spatial velocity potentials as follows:

$${\varphi }_1=\sum _{i=1}^2{e}^{\mathrm{i}{q}_{i}x}{\mathcal{Z}}_i(k_i,z)+\sum _{i=1}^{\infty }{R}_i{e}^{-\mathrm{i}{q}_{i}x}{\mathcal{Z}}_i(k_i,z),$$

(16)

$${\varphi }_2=\sum _{i=1}^{\infty }{T}_i{e}^{\mathrm{i}{q}_{i}x}{\mathcal{Z}}_i(k_i,z),$$

(17)

where the unknown coefficients $R_i$ and $T_i$ correspond to reflection and transmission, respectively. The eigenfunctions $I_i(k_i,z)$ in (10) are integrable in $(-H,0)$ and they form an orthogonal set in terms of the inner product as shown below:

$$〈{\mathcal{Z}}_i,{\mathcal{Z}}_j〉=\underset{-H}{\overset{-h}{\int }}{\mathcal{Z}}_i{\mathcal{Z}}_j\mathrm{d}z+\rho \underset{-h}{\overset{0}{\int }}{\mathcal{Z}}_i{\mathcal{Z}}_j\mathrm{d}z=(\begin{array}{cc}0,\hfill & i\ne j,\hfill \\ \multicolumn{1}{c}{W_i,}& i=j,\hfill \end{array}$$

(18)

where

$$\begin{array}{c}{W}_i=\frac{(k_i^2+{\mathcal{G}}^2)\sinh \left(2k_i{H}_1\right)+2\mathcal{G}{k}_i(1-\cosh \left(2k_i{H}_1\right))}{4k_i^3}\hfill \\ \multicolumn{1}{c}{+\frac{(2k_i^3-2{\mathcal{G}}^2{k}_i)H_1}{4k_1^3}+{\mathcal{Q}}_i[\frac{(k_i^2+K^2)\sinh \left(2k_{i}h\right)}{4k_i}}\\ \multicolumn{1}{c}{-\frac{(K^2-k_i^2)h+K(\cosh \left(2k_{i}h\right)-1)}{2}],}\end{array}$$

with

$${\mathcal{Q}}_i=\rho {\left(\frac{\sinh k_i{H}_1-\frac{\mathcal{G}}{k_i}\sinh k_i{H}_1}{K\cosh k_{i}h-k_i\sinh k_{i}h}\right)}^2.$$

We utilise the orthogonality of the eigenfunction in the continuity condition of velocity in (7), which yields

$$\begin{array}{c}{R}_i=1-T_i,\mathrm{for}i=1,2,\mathrm{and}{R}_i+T_i=0,\mathrm{for}i=3,4,\hfill \end{array}$$

(19)

Using the relation (19) for the second part of equation (6) and equality condition of ${\varphi }_1$ and ${\varphi }_2$ equation (7), we get

$$\sum _{i=1}^{\infty }{\alpha }_i{T}_i{\mathcal{Z}}_i=\beta \sum _{i=1}^2{\mathcal{Z}}_i,z\in (-a,0),$$

(20)

$$\sum _{i=1}^{\infty }{T}_i{\mathcal{Z}}_i=\sum _{i=1}^2{\mathcal{Z}}_i,z\in (-H,-a),$$

(21)

where ${\alpha }_i=(q_i+\beta )$ and $\beta =2k_1{\mathcal{G}}_1$ .

Dalrymple and Martin ⁷¹ used the least square approach to find the unknown coefficients in equations (20) and (21). For this task, a new quantity $S(z)$ with fixed evanescent modes $N$ is defined as follows:

$$S(z)\equiv \sum _{i=1}^{N+2}{S}_i^1(z)T_i-S_0(z),$$

(22)

where

$$\begin{array}{c}{S}_0(z)=\{\begin{array}{c}\beta \sum _{i=1}^2{\mathcal{Z}}_i,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^2{\mathcal{Z}}_i,\mathrm{for}z\in (-H,-a),}\end{array}\hfill \end{array}$$

$$S_i^1(z)=\{\begin{array}{c}{\alpha }_i{\mathcal{Z}}_i,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{{\mathcal{Z}}_i,\mathrm{for}z\in (-H,-a).}\end{array}$$

The following relationship must hold in order for this approach to work:

$$\underset{-H}{\overset{0}{\int }}|S(z){|}^2\mathrm{d}z\mathrm{be}\mathrm{minimized}.$$

(23)

Following this, we minimise equation (23) with regard to each and every $T_j$ , which results in

$$\underset{-H}{\overset{0}{\int }}\overline{S}(z)\frac{\partial S(z)}{\partial T_j}\mathrm{d}z=0,$$

(24)

with $\overline{S}$ denoting the complex conjugate of $S$ .

We arrive at the following system of linear equations after integration:

$$\sum _{i=1}^{N+2}{X}_{j,i}{\overline{T}}_i=Z_j,$$

(25)

where $X_{j,i}=\underset{-H}{\overset{0}{\int }}{\overline{S}}_i^1{S}_j^1\mathrm{d}z,Z_j=\underset{-H}{\overset{0}{\int }}{\overline{S}}_0{S}_j^1\mathrm{d}z$ .

Equation (25) yields a system of linear equations of order $(N+2)\times (N+2)$ written in compact form as

$$\mathbb{A}\mathbb{X}=\mathbb{B},\mathrm{with}\mathrm{the}\mathrm{unknown}\mathbb{X}={\left[\begin{array}{cccc}{\overline{T}}_1\hfill & {\overline{T}}_2\hfill & \dots \hfill & {\overline{T}}_{N+2}\hfill \end{array}\right]}^T.$$

(26)

Solving the above system of equations gives us ${\overline{T}}_i$ ’s, which can be used to obtain different physical values of interest. Using equation (19), the reflection coefficients $R_1$ and $R_2$ are calculated.

The free surface and interfacial reflection coefficients are denoted by ${\mathrm{R}}_{\mathrm{SM}}=|R_1|$ and ${\mathrm{R}}_{\mathrm{IM}}=|R_2|$ . The following formula is used to compute the non-dimensional horizontal wave force ${\mathrm{F}}_b$ acting on the perforated barrier:

$${\mathrm{F}}_b=\frac{\omega }{\mathrm{ghH}}{|\underset{-a}{\overset{-h}{\int }}[{\varphi }_2-{\varphi }_1]\mathrm{d}z+\rho \underset{-h}{\overset{0}{\int }}[{\varphi }_2-{\varphi }_1]\mathrm{d}z|}_{x=0}.$$

(27)

The amplitudes at the free surface and interface are obtained, respectively, as

$${\eta }_1=\frac{\mathrm{i}}{\omega }\frac{\partial {\varphi }_j(x,0)}{\partial z}\mathrm{and}{\eta }_2=\frac{\mathrm{i}}{\omega }\frac{\partial {\varphi }_j(x,-h)}{\partial z}\mathrm{for}j=1,2.$$

(28)

The non-dimensional pressure distribution at the surface of the perforated structure in each layer may be calculated by using the formula below:

$${\mathrm{D}}_P=\left|\frac{\mathrm{i}\Upsilon }{\omega \mathrm{hH}}({\varphi }_2(0,z)-{\varphi }_1(0,z))\right|,$$

(29)

where $\Upsilon =\{\begin{array}{c}\rho ,\mathrm{for}\mathrm{the}\mathrm{upper}\mathrm{layer}\mathrm{fluid},\hfill \\ \multicolumn{1}{c}{1,\mathrm{for}\mathrm{the}\mathrm{lower}\mathrm{layer}\mathrm{fluid}.}\end{array}$ .

Energy identity relation for scattering problem

In order to grasp the relationship between the hydrodynamic coefficients and subsequently to confirm the correctness and application of the computational results, energy identities must be constructed. We utilise Green’s integral theorem to develop energy identities.

As seen below, the whole fluid domain is separated into two bounded contours:

$${\tau }_1=\left(\begin{array}{c}x=-X,-h\le z\le 0;\hfill \\ \multicolumn{1}{c}{z=-h_{+},-X\le x\le 0;}\\ \multicolumn{1}{c}{z=-h_{-},-X\le x\le 0;}\\ \multicolumn{1}{c}{x=-X,-h\le z\le -H;}\\ \multicolumn{1}{c}{z=-H,-X\le x\le 0;}\\ \multicolumn{1}{c}{x=0,-H\le z\le -a}\\ \multicolumn{1}{c}{x=0,-a\le z\le -h;}\\ \multicolumn{1}{c}{x=0,-h\le z\le 0;}\\ \multicolumn{1}{c}{z=0,-X\le x\le 0,}\end{array}\right)$$

$${\tau }_2=\left(\begin{array}{c}x=0,-H\le z\le 0;\hfill \\ \multicolumn{1}{c}{x=0,-h\le z\le -a;}\\ \multicolumn{1}{c}{x=0,-H\le z\le -a;}\\ \multicolumn{1}{c}{z=-H,0\le x\le X;}\\ \multicolumn{1}{c}{x=X,-H\le z\le -h;}\\ \multicolumn{1}{c}{z=-h_{-},0\le x\le X}\\ \multicolumn{1}{c}{z=-h_{+},0\le x\le X;}\\ \multicolumn{1}{c}{x=X,-h\le z\le 0;}\\ \multicolumn{1}{c}{z=0,0\le x\le X,}\end{array}\right)$$

where the upper and lower layers are denoted $\mathrm{by}\pm$ , and the rightmost and leftmost boundaries are denoted by $x=\pm X$ , respectively.

Using Green’s integral theorem, we obtain

$$\underset{\tau }{\int }(\varphi \frac{\partial \overline{\varphi }}{\partial n}-\overline{\varphi }\frac{\partial \varphi }{\partial n})\mathrm{d}\tau =\sum _{j=1}^2\underset{{\tau }_j}{\int }({\varphi }_j\frac{\partial {\overline{\varphi }}_j}{\partial n}-{\overline{\varphi }}_j\frac{\partial {\varphi }_j}{\partial n})\mathrm{d}\tau =0.$$

(30)

The contribution of (30) along each line is determined by using the technique of Das et al. ⁷² At the interface, we obtain

$$\rho {({\varphi }_j\frac{\partial {\overline{\varphi }}_j}{\partial n}-{\overline{\varphi }}_j\frac{\partial {\varphi }_j}{\partial n})}_{z=-h_{+}}={({\varphi }_j\frac{\partial {\overline{\varphi }}_j}{\partial n}-{\overline{\varphi }}_j\frac{\partial {\varphi }_j}{\partial n})}_{z=-h_{-}}.$$

(31)

Using equation (31) in equation (30) gives rise to zero contribution along $z=-h_{\pm }$ .

By allowing $X\to \infty$ , the contribution from the line $x=\pm X$ gives

$${\mathcal{U}}_1=2\mathrm{i}\sum _{i=1}^2{q}_i(|R_i{|}^2-1)W_i,$$

(32)

$${\mathcal{U}}_2=2\mathrm{i}\sum _{i=1}^2{q}_i|T_i{|}^2{W}_i.$$

(33)

The contribution along $z=0$ and $z=-h$ will be 0 through the use of equations (3) and (4). The contribution from $x=0$ can be obtained as

$${\mathcal{U}}_3=\mathrm{i}{k}_1({\mathcal{G}}_1+{\overline{\mathcal{G}}}_1)[\rho \underset{-h}{\overset{0}{\int }}|{\varphi }_1-{\varphi }_2{|}^2\mathrm{d}z+\underset{-a}{\overset{-h}{\int }}|{\varphi }_1-{\varphi }_2{|}^2\mathrm{d}z].$$

(34)

Summing all the integrals, the final form of the energy balance relation is obtained as

$$\begin{array}{c}{\mathcal{U}}_1+{\mathcal{U}}_2+{\mathcal{U}}_3=0\hfill \\ \Rightarrow \sum _{i=1}^2{q}_i(|R_i{|}^2+|T_i{|}^2)W_i-\frac{\mathrm{i}}{2}{\mathcal{U}}_3=\sum _{i=1}^2{q}_i{W}_i.\hfill \end{array}$$

(35)

Let ${\mathrm{V}}_1=\sum _{i=1}^2{q}_i|R_i{|}^2{W}_i,{\mathrm{V}}_2=\sum _{i=1}^2{q}_i|T_i{|}^2{W}_i$ , ${\mathrm{V}}_3=-\frac{\mathrm{i}}{2}{\mathcal{U}}_3$ and ${\mathrm{V}}_4=\sum _{i=1}^2{q}_i{W}_i$ where $|V_1|$ is considered as reflected energy, $|V_2|$ as transmitted energy, $|V_3|$ the energy dissipated by the structure and $V_r=|V_4-V_1-V_2|$ .

The energy identity terms are compared to determine the validity of the solution. Table 1 clearly shows that the computational data are of a higher order of accuracy.

Table 1.

Verification of the computational data from energy identity terms with $h/H=0.5$ , $a/H=0.8$ .

$\mathcal{G}H$	$\mathrm{KH}$	${\mathcal{G}}_1$	$|V_1|$	$|V_2|$	$V_r$	$|V_3|$
$2$	$0.4$	$1$	$1.403$	$1.654$	$9.634$	$9.632$
		$1+\mathrm{i}$	$1.318$	$3.102$	$9.042$	$9.035$
	$0.63$	$1$	$11.634$	$3.351$	$12.287$	$12.284$
		$1+\mathrm{i}$	$10.42$	$5.851$	$11.002$	$11.005$
$10$	$0.4$	$1$	$3.583$	$19.875$	$16.877$	$16.772$
		$1+\mathrm{i}$	$2.413$	$26.56$	$11.351$	$11.292$
	$0.63$	$1$	$10.945$	$37.293$	$40.404$	$40.378$
		$1+\mathrm{i}$	$7.708$	$52.481$	$28.453$	$28.434$

Trapping problem

Method of solution

For the trapping problem, the only change will occur in ${\varphi }_2$ due to the rigid wall condition given by equation (8). Elsewhere, potential ${\varphi }_1$ remains the same as earlier. We can express ${\varphi }_2$ as

$${\varphi }_2=\sum _{i=1}^{\infty }{T}_i^{(r)}\cos q_i(x-L){\mathcal{Z}}_i(k_i,z).$$

(36)

Now, utilising the orthogonality of ${\mathcal{Z}}_i$ in (7), we obtain

$$\begin{array}{c}1=R_i-\mathrm{i}{T}_i^{(r)}\sin q_{i}L,\mathrm{for}i=1,2,\mathrm{and}\hfill \\ \multicolumn{1}{c}{R_i-\mathrm{i}{T}_i^{(r)}\sin q_{i}L=0,\mathrm{for}i=3,4,}\end{array}$$

(37)

Analysing (37), we can conclude that

$$\begin{array}{c}{R}_i=1\mathrm{for}n=1,2,\hfill \\ \multicolumn{1}{c}{R_i=0\mathrm{for}n=3,4,\dots ,}\end{array}\}\mathrm{for}L=\frac{m\pi }{q_i},\mathrm{with}m=1,2,\dots .$$

(38)

Moreover, for a normal incident wave, $\theta =0^{°}$ which results in $L=\frac{m{\lambda }_i}{2}$ for $i=1,2$ with $\lambda =\frac{2\pi }{k_i}$ . It clearly shows that, for normal incident waves, the perforated barrier ensures maximum reflection for $\frac{m{\lambda }_i}{2}$ positions with integer $m$ . Now, using the relation (37) for the second part of equation (6) and equality condition of ${\varphi }_1$ and ${\varphi }_2$ (equation (7)), we get

$$\sum _{i=1}^{\infty }(\beta {\hat{\alpha }}_i-\mathrm{i}{q}_i\sin q_{i}L)T_i^{(r)}{\mathcal{Z}}_i=\beta \sum _{i=1}^2{\mathcal{Z}}_i,z\in (-a,0),$$

(39)

$$\sum _{i=1}^{\infty }{\hat{\alpha }}_i{T}_i^{(r)}{\mathcal{Z}}_i=\sum _{i=1}^2{\mathcal{Z}}_i,z\in (-H,-a),$$

(40)

where ${\hat{\alpha }}_i=\frac{(\cos q_{i}L-\mathrm{i}\sin q_{i}L)}{2}$ .

A new quantity $\mathbb{S}(z)$ with fixed evanescent modes $N$ is defined as follows:

$$\mathbb{S}(z)\equiv \sum _{i=1}^{N+2}{\mathbb{S}}_i^1(z)T_i^{(r)}-{\mathbb{S}}_0(z),$$

(41)

where

$${\mathbb{S}}_0(z)=\{\begin{array}{c}\beta \sum _{i=1}^2{\mathcal{Z}}_i,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^2{\mathcal{Z}}_i,\mathrm{for}z\in (-H,-a),}\end{array}$$

$${\mathbb{S}}_i^1(z)=\{\begin{array}{c}(\beta {\hat{\alpha }}_i-\mathrm{i}{q}_i\sin q_{i}L){\mathcal{Z}}_i,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{{\hat{\alpha }}_i{\mathcal{Z}}_i,\mathrm{for}z\in (-H,-a).}\end{array}$$

The following system of linear equations results from integration with regard to $z$ :

$$\sum _{i=1}^{N+2}{\hat{X}}_{j,i}{\overline{T}}_i^{(r)}={\hat{Z}}_j,$$

(42)

where ${\hat{X}}_{j,i}=\underset{-H}{\overset{0}{\int }}{\overline{\mathbb{S}}}_i^1{\mathbb{S}}_j^1\mathrm{d}z,{\hat{Z}}_j=\underset{-H}{\overset{0}{\int }}{\overline{\mathbb{S}}}_0{\mathbb{S}}_j^1\mathrm{d}z$ .

Equation (42) yields a system of linear equations of order $(N+2)\times (N+2)$ as

$$\mathcal{A}\mathfrak{X}=\mathcal{B},\mathrm{with}\mathrm{the}\mathrm{unknown}\mathfrak{X}={\left[\begin{array}{cccc}{\overline{T}}_1^{(r)}\hfill & {\overline{T}}_2^{(r)}\hfill & \dots \hfill & {\overline{T}}_{N+2}^{(r)}\hfill \end{array}\right]}^T.$$

(43)

Solving the above system of equations gives the unknowns, which can be used to obtain different physical values of interest. The following formula is used to compute the non-dimensional horizontal wave force ${\mathrm{F}}_w$ operating on the rigid wall:

$${\mathrm{F}}_w=\frac{\omega }{\mathrm{ghH}}\underset{-H}{\overset{-h}{\int }}{\varphi }_2(L,z)\mathrm{d}z+\rho \underset{-h}{\overset{0}{\int }}{\varphi }_2(L,z)\mathrm{d}z.$$

(44)

Energy identity relation for trapping problem

The contribution of ${\mathcal{U}}_2$ will be 0 due to the rigid wall condition. The energy balance relation for the trapping model is as follows:

$$\begin{array}{c}{\mathcal{U}}_1+{\mathcal{U}}_3=0\hfill \\ \Rightarrow \sum _{i=1}^2{q}_i|R_i{|}^2{W}_i-\frac{\mathrm{i}}{2}{\mathcal{U}}_3=\sum _{i=1}^2{q}_i{W}_i.\hfill \end{array}$$

(45)

Special case: Radiation problem due to perforated barrier

There are six motions that a floating structure may produce: (i) three translational motions known as surge, heave and sway along the $x$ -, $y$ - and $z$ -axes, respectively and (ii) three rotational motions known as roll, pitch and yaw about the $x$ -, $y$ - and $z$ -axes, respectively. When the perforated structure is assumed to be in motion, the situation raises the possibility of radiation problems. Figure 3 shows the horizontal oscillation of the structure about the $x$ -axis with an angular frequency $\omega$ and slight displacement $a$ . In this case, a mechanical or energy converter or structure is what we may expect the barrier to be. The motion of the barrier generates outgoing water waves. Considering the thin structural design, we are only interested in the surge motion.

Figure 3.

Schematic diagram of radiation problem.

Mathematical formulation and method of solution

The radiation problem of the thin barrier is based on considering the small motion of the thin barrier. Similar consideration can be observed in Lee and Chwang. ⁸ Why only the surge motion is taken into account in this arrangement and not also the heave motion, the following explanation is provided for justification:

Because the barrier is thin, the waves interacting with it are negligible from the $z$ -direction. Waves usually interact with vertical floating structures only if they have a firm foundation. With this in mind, heave motion for this structure is not considered since its impact would be negligible. We express the spatial radiation velocity potentials ${\varphi }_m^R$ for $m=1,2$ in the following forms:

$${\varphi }_1^R=\sum _{i=1}^{\infty }{\mathcal{A}}_i{e}^{-\mathrm{i}{q}_{i}x}{\mathcal{Z}}_i(k_i,z),$$

(46)

$${\varphi }_2^R=\sum _{i=1}^{\infty }{\mathcal{B}}_i{e}^{\mathrm{i}{q}_{i}x}{\mathcal{Z}}_i(k_i,z),$$

(47)

where the coefficients ${\mathcal{A}}_i$ and ${\mathcal{B}}_i$ are the coefficients of the radiated potentials.

The perforated structure has horizontal velocity $U(z,t)=-\mathrm{i}\omega u_0{e}^{-\mathrm{i}\omega t}$ , which generates outgoing waves on both sides. All other boundary conditions are still valid except equation (6), which, in this case, is replaced by

$$\frac{\partial {\varphi }_1^R}{\partial x}=\frac{\partial {\varphi }_2^R}{\partial x}=-\mathrm{i}\omega u_0+\mathrm{i}{k}_1{\mathcal{G}}_1({\varphi }_1^R-{\varphi }_2^R).$$

(48)

Using the orthogonality of depth function in $\frac{\partial {\varphi }_1^R}{\partial x}=\frac{\partial {\varphi }_2^R}{\partial x}$ , we obtain

$${\mathcal{A}}_i=-{\mathcal{B}}_i.$$

(49)

Using relation (49) in equation (48) and equality condition of ${\varphi }_1^R$ and ${\varphi }_2^R$ (equation (7)), we get

$$\sum _{i=1}^{\infty }{\alpha }_i{\mathcal{B}}_i{\mathcal{Z}}_i=-\omega u_0,z\in (-a,0),$$

(50)

$$\sum _{i=1}^{\infty }{\mathcal{B}}_i{\mathcal{Z}}_i=0,z\in (-H,-a).$$

(51)

Following a similar approach, $\mathcal{S}(z)$ is defined as

$$\mathcal{S}(z)\equiv \sum _{i=1}^{N+2}{\mathcal{S}}_i^1(z){\mathcal{B}}_i-{\mathcal{S}}_0(z),$$

(52)

where

$${\mathcal{S}}_0(z)=\{\begin{array}{c}-u_0\omega ,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{0,\mathrm{for}z\in (-H,-a),}\end{array},$$

$${\mathcal{S}}_i^1(z)=\{\begin{array}{c}{\alpha }_i{\mathcal{Z}}_i,\mathrm{for}z\in (-a,0),\hfill \\ \multicolumn{1}{c}{{\mathcal{Z}}_i,\mathrm{for}z\in (-H,-a).}\end{array}$$

Integrating with respect to each ${\mathcal{B}}_j$ gives the following system of linear equations:

$$\sum _{j=1}^{N+2}{\mathcal{X}}_{i,j}{\overline{\mathcal{B}}}_j={\hat{\mathcal{Z}}}_i,\mathrm{for}i=1,2,\dots ,N+2,$$

(53)

where ${\mathcal{X}}_{i,j}=\underset{-H}{\overset{0}{\int }}{\overline{\mathcal{S}}}_j^1{\mathcal{S}}_i^1\mathrm{d}z,{\hat{\mathcal{Z}}}_i=\underset{-H}{\overset{0}{\int }}{\overline{\mathcal{S}}}_0{\mathcal{S}}_i^1\mathrm{d}z$ .

Equation (53) can be written equivalently as a linear system of complex matrix system of size $(N+2)\times (N+2)$ as below:

$$\hat{\mathbb{A}}\hat{\mathbb{X}}=\hat{\mathbb{B}}\mathrm{with}\hat{\mathbb{X}}={\left[\begin{array}{cccc}{\overline{\mathcal{B}}}_1\hfill & {\overline{\mathcal{B}}}_2\hfill & \dots \hfill & {\overline{\mathcal{B}}}_{N+2}\hfill \end{array}\right]}^T.$$

(54)

The amplitude to stroke ratio, that is, the ratio of the amplitude of the radiated potential to the amplitude of barrier stroke in both propagating modes, can be defined by ${\mathrm{R}}_1^a=|{\mathcal{A}}_1/u_0|$ and ${\mathrm{R}}_2^a=|{\mathcal{A}}_2/u_0|$ , respectively.

Solving the radiation problem yields key hydrodynamic coefficients such as added mass and damping coefficients. These coefficients are derived from the real and imaginary components of the hydrodynamic response loads on the body induced by the specified body motions. Added mass refers to the weight added to a system in a fluid as a result of an accelerating or decelerating entity having to transfer some volume of surrounding fluid with it as it travels. Damping is a phenomenon that decreases, limits, or stops oscillations within or on an oscillatory system. The hydrodynamic forces in the $x$ -direction (i.e. for surge motion) caused by the motion may be calculated by integrating the related pressure throughout the structure. The non-dimensional form of the added mass coefficient $(\mu )$ and the damping coefficient $(\lambda )$ may also be obtained using the approach of Hsu and Wu ⁷³ :

$$\begin{array}{c}\mu =\frac{\sqrt{\mathrm{gH}}}{u_0\omega }(\underset{-a}{\overset{-h}{\int }}\mathrm{Im}({\varphi }_1^R-{\varphi }_2^R)\mathrm{d}z\hfill \\ {+\rho \underset{-h}{\overset{0}{\int }}\mathrm{Im}({\varphi }_1^R-{\varphi }_2^R)\mathrm{d}z)}_{x=0},\hfill \end{array}$$

(55)

$$\begin{array}{c}\frac{H}{\lambda =u_0}(\underset{-a}{\overset{-h}{\int }}\mathrm{Re}({\varphi }_1^R-{\varphi }_2^R)\mathrm{d}z\hfill \\ {+\rho \underset{-h}{\overset{0}{\int }}\mathrm{Re}({\varphi }_1^R-{\varphi }_2^R)\mathrm{d}z)}_{x=0}.\hfill \end{array}$$

(56)

Numerical results and discussion

We numerically compute the solution of (26) by using Matlab R2019a with some parameter values as follows: $h/H=0.5,\rho =0.9$ and $\mathrm{KH}=0.23$ . These values are kept fixed unless otherwise mentioned.

Convergence of evanescent modes

The number of evanescent modes $(N)$ needs to be fixed for the accuracy of the solution. In comparison to the scenario when evanescent modes are not considered (i.e. $N=0$ ), Table 2 indicates a significant contribution of the evanescent modes to reflection coefficients which will further affect the energy contributions. The values of two successive reflection coefficients are almost identical as $N$ grows, which ensures the convergence of the evanescent modes. As a result, the number of evanescent modes is estimated to be $N=16$ , which may be considered a reasonable approximation for our further $\mathrm{calculation}$ , as shown in Table 2.

Table 2.

Convergence study of ${\mathrm{R}}_{\mathrm{SM}}$ and ${\mathrm{R}}_{\mathrm{IM}}$ with $a/H=0.7,\theta =30°$ and $\mathcal{G}H=10$ .

$N$	${\mathrm{R}}_{\mathrm{SM}}$	${\mathrm{R}}_{\mathrm{IM}}$
$0$	$0.293$	$0.113$
$4$	$0.292$	$0.118$
$8$	$0.291$	$0.112$
$12$	$0.291$	$0.109$
$16$	$0.291$	$0.108$
$18$	$0.291$	$0.108$
$20$	$0.291$	$0.108$

Scattering problem

Effect of porous-effect parameter of the barrier on reflection coefficients

Ideal structural porosity minimises the fluid passing across the barrier, and the damping effect minimises the waveload on the shoreline. The impact of porous-effect parameter ${\mathcal{G}}_1$ on hydrodynamic coefficients concerning the incident angle $\theta$ is illustrated in Figure 4. The reflection coefficients exhibit similar behaviour when ${\mathcal{G}}_1$ is varied. Higher amplitude variations in both propagation modes are attained for ${\mathcal{G}}_1=0$ , which is due to the higher reflection of waves by the rigid structure. Both the reflection coefficients decrease as ${\mathcal{G}}_1$ increases. This clearly shows the damping effect, which gives rise to enhanced wave decay. For Figure 4(a), regardless of the value of the porous-effect parameter, maximum reflection is seen for both propagating modes at $\theta \approx 0^{°}$ , and zero minimum is attained at ${90}^{°}$ . Reflection coefficients decrease with increasing $\theta$ , which is similar to the studies in a homogeneous fluid of Sahoo et al. ¹⁰ with a thin perforated barrier and Losada et al. ⁷⁴ with a thin and rigid barrier. For Figure 4(b), maximum reflection is seen for both propagating modes at $\theta \approx {90}^{°}$ , and a kink minimum is observed near ${70}^{°}$ , which is also observed by Dalrymple et al. ⁵ The kink minimum for the reflection coefficient is missing in our case for $\mathcal{G}H=10$ (Figure 4(a)), which may be due to the higher bottom permeability parameter for which the resistance impact of the barrier is more dominant than the inertial impact. For the thick porous structure model (See Sollitt and Cross ¹ and Dalrymple et al. ⁵ ), the inertial effect is assumed to be the reason for such behaviour. ¹⁰

Figure 4.

Impact of bottom permeability (a) $\mathcal{G}H=10$ , (b) $\mathcal{G}H=5$ on non-dimensional reflection coefficients against non-dimensional incident angle $\theta$ corresponding to various structural porous-effect parameter values with $\mathrm{KH}=0.2,a/H=0.7$ .

Effect of density stratification on reflection coefficients while varying frequency

Figure 5(a) and (b) show the effect of density stratification on hydrodynamic coefficients against the incoming wave frequency. When $\mathrm{KH}$ is changed, the reflection coefficients show similar behaviour. Higher frequency waves result in increased amplitude fluctuations in both propagation modes, which is related to higher wave reflection. For $\mathrm{KH}\le 1$ , overall free surface reflection is more significant than interfacial reflection. While $\rho \to 1$ , the bottom permeability $\mathcal{G}H$ has a crucial impact. For $\mathcal{G}H=2$ , a region with higher interfacial wave reflection is observed (Figure 5(a)) whereas free surface reflection is observed to be consistently greater in Figure 5(b). This demonstrates that the influence of $\mathcal{G}H$ is maximal for high-frequency waves by higher dissipating interfacial waves. Moreover, this also demonstrates that density stratification can be seen more clearly for high-frequency waves. The highest reflection is found for free surface propagating mode, regardless of the value of the permeability.

Figure 5.

Non-dimensional reflection coefficients versus $\mathrm{KH}$ and $\rho$ for (a) $\mathcal{G}H=2$ and (b) $\mathcal{G}H=10$ with $a/H=0.8$ , ${\mathcal{G}}_1=1+\mathrm{i}$ and $\theta ={45}^{°}$ .

Comparison with homogeneous fluid model

We approximate our model for a homogeneous fluid framework. For Figure 6(a) and (b), we consider $h/H={10}^{-3},\rho =0.99,\theta =0^{°},\mathcal{G}=0,\mathrm{KH}=0.1$ (Figure 6(a)) and $a/H=0.5$ (Figure 6(b)). Such a consideration makes our model resemble with that of Lee and Chwang. ⁸ Due to the lack of availability of the numerical data, we plot our results for $\mathrm{KH}=0.4$ (Figure 6(a)). Both the figures of Figure 6 show a similar pattern to those in Lee and Chwang. ⁸ Figure 6(a) shows that increasing the barrier porous-effect parameter has less effect on the reflection coefficient, which is the same as in Lee and Chwang. ⁸ However, a minimum near $a/H\approx 1$ can be observed. This change can be attributed to density and frequency differences in the two layers. In addition, Figure 6(b) shows a good comparison for the dimensionless waveload and both the works show that increased ${\mathcal{G}}_1$ results in less waveload. For Figure 6(c), the same numerical values are considered except $\mathrm{KH}=1$ and $\theta ={45}^{°}$ . Figure 6(c) and Fig. 8 of Sahoo et al. ¹⁰ show similar pattern and characteristics except near $a/H\approx 1$ . Overall, the graphs which represent an approximation of our two-layer model to a homogeneous fluid model shows a similar pattern observed in the examples as above for the homogeneous fluid model which adds additional support of verification of our computed results.

Figure 6.

(a) ${\mathrm{R}}_{\mathrm{SM}}$ versus $a/H$ , (b) ${\mathrm{F}}_b$ versus $k_{1}H$ and (c) ${\mathrm{R}}_{\mathrm{SM}}$ versus $1-\frac{a}{H}$ corresponding to various values of ${\mathcal{G}}_1$ in homogeneous fluid model approximation with $a/H=0.5$ .

Effect of porous-effect parameter of the barrier on waveload and pressure

For increasing angular frequency, the influence of the porous-effect parameter on waveload ${\mathrm{F}}_b$ is clearly observed in Figure 7(a). As ${\mathcal{G}}_1$ rises, the effective waveload ${\mathrm{F}}_b$ on the structure decreases. From a physical aspect, significant wave absorption is possible due to viscous damping via pore gaps, limiting the number of waves and causing the waveload to behave this way. Lower frequency waves present the best waveload on the structure because the amount of wave interaction increases as the wave period lengthens. From Figure 7(b), we can observe similar nature of different pressure distribution curves when ${\mathcal{G}}_1$ is varied. Because of the effective waveload characteristics, when the absolute amount of ${\mathcal{G}}_1$ rises, the average hydrodynamic pressure ${\mathrm{D}}_P$ gets lowered. The hydrodynamic pressure ${\mathrm{D}}_P$ at the free surface is also greater than that at the bottom. The interface has a narrower pressure distribution. As the porosity rises, the pressure minima in pressure distribution shift towards the barrier end edge, indicating a wave energy shift owing to the existence of the barrier.

Figure 7.

Non-dimensional waveload on the structure (a) ${\mathrm{F}}_b$ versus $\mathrm{KH}$ and non-dimensional pressure distribution on the structure (b) ${\mathrm{D}}_P$ versus $z/H$ for various ${\mathcal{G}}_1$ with $a/H=0.7,\theta ={45}^{°}$ and $\mathcal{G}H=10$ .

Effect of the porous-effect parameter of the barrier on the free surface and interfacial elevations

Figure 8(a) and (b) demonstrate how various values of the porous-effect parameter ${\mathcal{G}}_1$ affect the free surface and interfacial elevation. In both propagating modes, the existence of the perforated structure leads to a significant drop in elevation since the wave energy dissipation is larger in the presence of a surface-piercing barrier than in the absence of one. The interfacial wave amplitude ${\eta }_2/H$ , on the other hand, is greater than ${\eta }_1/H$ , as shown in Figure 8(b). We may depict the resonating interaction of waves in the free surface and interface as the reason for such characteristics. However, the dead-water analogue is also important to this observation. In both propagating modes, the oscillation amplitude for $x\le 0$ is lower when the absolute value of ${\mathcal{G}}_1$ is increased. This occurs because, as shown in Figure 8(a), an effective waveload for the incoming wave energy is lower when the value of the porous-effect parameter is raised. The major observation for these graphs is the discontinuity between graphs at $x=0$ . Moreover, the difference of elevation amplitudes at both sides of the barrier decreases for increasing ${\mathcal{G}}_1$ . Such a result was also observed by Sahoo et al. ¹⁰ and Ursell. ⁷⁵ This may be due to the structural dissipation, which can be attributed largely to the thick porous structure model Barman and Bora. ²² Moreover, we can observe that the phase shift of the optimum value of elevations also varies as the absolute value of the porous-effect parameter ${\mathcal{G}}_1$ changes. Real ${\mathcal{G}}_1$ values show no significant changes, whereas complex ${\mathcal{G}}_1$ values show a significant difference since the complex porous-effect parameter allows inertial damping. For both propagating modes, we can observe opposite patterns at both sides of $x=0$ while varying ${\mathcal{G}}_1$ . This may be due to the wave blockage for barrier porosity which is higher for a rigid barrier.

Figure 8.

(a) ${\eta }_1/H$ versus $x/H$ and (b) ${\eta }_2/H$ versus $x/H$ for different ${\mathcal{G}}_1$ with $a/H=0.7,\theta =45°$ and $\mathrm{GH}=10$ .

Effect of porous-effect parameter on the energy contribution of the barrier

Figure 9(a) and (b) show the variation in $|{\mathrm{V}}_3|$ versus $\mathcal{G}H$ for different values of $\mathcal{G}H$ . The wave energy dissipation attains a minimum after $\mathcal{G}H=5$ . Figure 9 also helps in finding out the major contribution of the bottom permeability in energy dissipation. The porosity of the vertical barrier helps to dissipate wave energy, as well as a major portion of the wave gets reflected and transmitted due to the presence of the structure. Initially, ${\mathcal{G}}_1=2$ gives the highest energy contribution, and increasing bottom permeability lowers the energy distribution irrespective the values of $\mathrm{KH}$ . Moreover, the figures depict that the real ${\mathcal{G}}_1$ is more effective for permeable bottom profiles with lower $|{\mathcal{G}}_1|$ . This may be because the barrier blocks waves more than it dissipates energy at lower bottom profiles. Figure 9 further indicates that the energy contribution gets lowered for increasing values of $\mathcal{G}H$ , and the minimum of $|V_3|$ also gets shifted while changing the frequency $\mathrm{KH}$ . Such points of minimum can be termed as the cut-off value of $\mathcal{G}$ , and after that, the pattern of graphs remains the same when the porous-effect parameter ${\mathcal{G}}_1$ is varied. The energy contribution from reflection and transmission is significant for these cut-off points. For higher values of $\mathcal{G}H$ , it is considered that higher $|{\mathcal{G}}_1|$ gives a suitable porous-effect parameter value of the plate for attaining higher reflection. Depending upon wave frequency, we can conclude that, for lower bottom permeability, the resistance effect of the barrier is major, whereas, for higher bottom permeability, the inertial effect of the barrier plays a crucial role in energy contribution.

Figure 9.

$|V_3|$ versus $\mathcal{G}H$ for different ${\mathcal{G}}_1$ with (a) $\mathrm{KH}=0.3$ and (b) $\mathrm{KH}=0.2$ , $\theta ={45}^{°}$ and $a/H=0.8$ .

Trapping problem

Effect of ${\mathcal{G}}_1$ on reflection coefficients

In Figure 10(a) and (b), the influence of the porous-effect parameter ${\mathcal{G}}_1$ on reflection coefficients is studied while varying $L/{\lambda }_i$ . Reflection coefficients in both propagating modes exhibit periodic behaviour for varying $L/{\lambda }_i$ . The periodic minimum of reflection coefficients in that confined region is defined as wave trapping Behera et al. ¹² For both propagating modes, the minimum for real ${\mathcal{G}}_1$ follows the same pattern while the minimum for complex ${\mathcal{G}}_1$ follows a different pattern. We can observe the variation of trapped location in both propagating modes while changing ${\mathcal{G}}_1$ , which is due to the shift in the phase angle while varying ${\mathcal{G}}_1$ . This also demonstrates a major contribution of complex ${\mathcal{G}}_1$ . As the porous-effect parameter ${\mathcal{G}}_1$ increases from 0 to 1, both ${\mathrm{R}}_{\mathrm{SM}}$ and ${\mathrm{R}}_{\mathrm{IM}}$ increase at first. But change in ${\mathrm{R}}_{\mathrm{SM}}$ or ${\mathrm{R}}_{\mathrm{IM}}$ follows opposite characteristics while increasing ${\mathcal{G}}_1$ from ${\mathcal{G}}_1=1$ . As a result, the barrier traps fewer waves for $|{\mathcal{G}}_1|>1$ than for $|{\mathcal{G}}_1|\le 1$ . Sahoo et al., ⁹ for ${\mathcal{G}}_1\ge 1$ , noticed a similar trend. The change of minimum value for each graph indicates that the inertial effect (the imaginary component of ${\mathcal{G}}_1$ ) is a key factor in wave trapping which diminishes as ${\mathcal{G}}_1$ increases. In addition, the influence of $\mathcal{G}H$ is shown in Figure 10(c) and (d). Higher ${\mathcal{G}}_1$ reduces reflection in both propagating modes due to structural dissipation. It is worth noting that the minimum for ${\mathrm{R}}_{\mathrm{SM}}$ occur at the same locations for various ${\mathcal{G}}_1$ of the barrier. However, the minimum of ${\mathrm{R}}_{\mathrm{IM}}$ shifts while varying ${\mathcal{G}}_1$ for $\mathcal{G}H\le 8$ . For ${\mathrm{R}}_{\mathrm{IM}}$ , real ${\mathcal{G}}_1$ yields a similar pattern, but complex ${\mathcal{G}}_1$ yields a distinct pattern. Reflection coefficients become stable as $\mathcal{G}H$ increases. This clearly illustrates that bottom permeability significantly influences wave dissipation up to a certain value. Furthermore, the value of $\mathcal{G}H$ , for which steadiness can be observed, is higher for interfacial waves than the free surface wave. This may be due to the major contribution of bottom porosity in dissipating interfacial waves. However, the influence of bottom permeability in the trapping model seems minimal beyond a certain value of $\mathcal{G}H$ .

Figure 10.

Impact of ${\mathcal{G}}_1$ on ${\mathrm{R}}_{\mathrm{SM}}$ and ${\mathrm{R}}_{\mathrm{IM}}$ against (a and b) $L/{\lambda }_i$ and (c and d) $\mathcal{G}H$ with $\theta =30°,\rho =0.7,\mathrm{KH}=0.75,a/H=0.7$ , $\mathcal{G}H=3$ (a and b) and $L/H=1$ (c and d).

Comparison with a homogeneous fluid model

We approximate our model in a homogeneous fluid framework by considering $h/H={10}^{-3},\rho =0.99,\theta =0^{°},\mathcal{G}=0,\mathrm{KH}=1$ . Such consideration makes our model resemble with that of Sahoo et al. ⁹ Figure 11 here and Fig. 3 of Sahoo et al. ⁹ show the same graphs for different $a/H$ . Similar nature of $R_{\mathrm{SM}}$ can be observed which shows increase of the length $(a/H)$ reduces overall reflection. It clearly verifies the computed results for the trapping model.

Figure 11.

$R_{\mathrm{SM}}$ versus ${\mathcal{G}}_1$ corresponding to various values of $a/H$ in homogeneous fluid approximation.

Effect of porous-effect parameter of the barrier on waveload

The impact of the porous-effect parameter ${\mathcal{G}}_1$ on waveload ${\mathrm{F}}_w$ while changing $L/H$ is shown in Figure 12. The effective waveload ${\mathrm{F}}_w$ on the wall shows a similar periodic nature as observed in Figure 10, and ${\mathrm{F}}_w$ increases as ${\mathcal{G}}_1$ increases. Due to higher wave flow through pore gaps, the waveload behaves in this fashion. While changing $\mathcal{G}H$ , we can clearly observe a similar periodic pattern. When $\mathcal{G}H$ increases, the average hydrodynamic waveload ${\mathrm{F}}_w$ decreases due to the higher dissipation by the bottom. There is no major change in the location of the minima while changing ${\mathcal{G}}_1$ . However, a small shift can be observed for complex ${\mathcal{G}}_1$ .

Figure 12.

${\mathrm{F}}_w$ versus $L/H$ for different ${\mathcal{G}}_1$ with (a) $\mathcal{G}H=10$ and (b) $\mathcal{G}H=6$ , $a/H=0.7$ and $\theta =30°$ .

Radiation problem

Numerical validation of the barrier model

We compare our present barrier model to a prior one by eliminating the permeability of the sea-bed from the formulation so as to compare with the existing wavemaker model of Manam and Sahoo. ¹¹ To convert the permeable bottom into an impermeable one, $\mathcal{G}$ is changed to $0$ , and a fully submerged structure is considered by assuming $a=H$ . The main difference is the different vertical matching boundary conditions due to the wavemaker and ‘no evanscent modes’. ¹¹ Furthermore, they defined the effective waveload on a wavemaker in the following way as we have defined for the scattering model:

$${\mathrm{F}}_b=\frac{K^2}{u_0}{|\underset{-a}{\overset{-h}{\int }}[{\varphi }_2^R-{\varphi }_1^R]\mathrm{d}z+\rho \underset{-h}{\overset{0}{\int }}[{\varphi }_2^R-{\varphi }_1^R]\mathrm{d}z|}_{x=0}.$$

(57)

In such a scenario, we use the following parameter values to approximate a comparison: $N=0,h/H=0.5,a/H=1,\theta ={60}^{°},{\mathcal{G}}_1=1+0.5\mathrm{i}$ and $u_0=1$ . The plots of the non-dimensional waveload ( ${\mathrm{F}}_b$ ) versus $k_{1}H$ for different density ratio $(\rho )$ are presented in Figure 13, and a comparison with Fig. 4 of Manam and Sahoo ¹¹ demonstrates a high degree of consistency. This confirms a higher accuracy of the computed results of the present model and its potential applicability to a wide range of issues related to marine infrastructure.

Figure 13.

Non-dimensional waveload ${\mathrm{F}}_{\mathrm{b}}$ corresponding to $k_{1}H$ for various density ratio $\rho$ .

Effect of ${\mathcal{G}}_1$ on radiated wave amplitude stroke ratio

The efficiency of the barrier is evaluated by comparing the amplitude to stroke ratio for various values of the porous-effect parameter ${\mathcal{G}}_1$ to the dimensionless angular frequency $\mathrm{KH}$ (Figure 14). Both ${\mathrm{R}}_1^{\mathrm{a}}$ and ${\mathrm{R}}_2^{\mathrm{a}}$ show a similar pattern, and a higher amplitude stroke ratio can be observed in both propagating modes for initial frequency values. Moreover, the amplitude ratio vanishes for a higher frequency of the barrier in both propagating modes. However, for the free surface mode, it can be observed for higher values of $\mathrm{KH}\approx 7$ , whereas for the interfacial mode, it can be observed after $\mathrm{KH}=0.5$ . Moreover, the amplitude ratio on the free surface is higher than on the interface. This shows that the impact of radiated wave energy is mainly carried out along the free surface waves. Consideration of different ${\mathcal{G}}_1$ results in significant differences in radiation coefficients, while a perforated barrier with lower ${\mathcal{G}}_1$ produces higher wave thrust as the porous effect damps the wave. Furthermore, a higher bottom permeability results in less amplitude stroke ratio for both propagating modes, which clearly shows the considerable impact of the wave dissipation by the bottom. Amplitude stroke ratio patterns for both modes show a similar nature. For lower $\mathcal{G}H$ , the frequency for obtaining maximum wave thrust is the same for various ${\mathcal{G}}_1$ . However, we can observe a shift in $\mathrm{KH}$ while obtaining a maximum higher thrust for various ${\mathcal{G}}_1$ . It clearly shows that the bottom permeability shifts the phase of the radiated waves. Moreover, the characteristics of ${\mathrm{R}}_1^{\mathrm{a}}$ for various ${\mathcal{G}}_1$ are of opposite nature before attaining the maximum value. However, no such behaviour can be observed for ${\mathrm{R}}_2^{\mathrm{a}}$ .

Figure 14.

Amplitude of stroke ratio for radiated potentials for (a) free surface mode and (b) interfacial mode versus $\mathrm{KH}$ different values of ${\mathcal{G}}_1$ for $a/H=0.7,u_0/H=0.2$ .

Effect of ${\mathcal{G}}_1$ on added mass and damping coefficient

For the barrier, added mass and damping coefficient vary against the dimensionless angular frequency $\mathrm{KH}$ for different porous-effect parameters (Figure 15). Added mass and damping coefficient increase sharply with the increase of oscillation amplitude. After $\mathrm{KH}\approx 3$ , an increase of oscillation frequency smoothly lowers added mass. Similar characteristics can be observed for the damping coefficient. A structure with the lowest porous-effect parameter yields the highest wave elevation for both cases. Usually, we observe that a rigid barrier produces the highest pressure distribution. Therefore, it will give a higher wave thrust, resulting in higher amplitude for radiated potentials. However, comparing the real and complex porous-effect parameters of the barrier, the complex parameter produces higher added mass. This is due to the increasing inertial effect of the barrier. Increasing porosity decreases the oscillation for the damping coefficient, which is similar to what is observed in Figure 14.

Figure 15.

(a) Added mass and (b) damping coefficient versus $\mathrm{KH}$ different values of ${\mathcal{G}}_1$ for $a/H=0.8,u_0/H=1$ and $\mathcal{G}H=1$ .

Conclusion

The current research focuses on the interaction of oblique water waves with a perforated barrier in a two-layer fluid flowing over a permeable bottom. The solution to the relevant scattering, trapping and radiation problem is found by using matched eigenfunction technique along with the least square approach. Mathematically, the bottom permeability appears as embedded in the propagating wave modes and the velocity potentials, and consequently, the effect can be observed in dead water analogue and energy identity terms. The physical impact of the bottom can be clearly observed in the hydrodynamic coefficients. Geometrically, the fluid region is divided into a number of sub-regions, and the specified equations and matching conditions are used to create a system of linear algebraic equations that are solved to get and evaluate certain vital conclusions, which are listed below:

• By selecting suitable structural porous-effect parameters, it is possible to obtain the maximum dissipation of energy and lower elevation than for the case of a similar rigid barrier. Higher damping of the perforated barrier results in lower wave elevation for both propagating modes. Moreover, higher ${\mathcal{G}}_1$ shifts the minima of $D_P$ towards the lower edge.

• From energy dissipation results for various porous-effect parameters, the following observations may be noted: (i) the barrier produces higher energy contribution for lower $|{\mathcal{G}}_1|$ values depending upon bottom permeability, (ii) the optimum energy dissipation shifts corresponding to changes in the wave frequency.

• Study of the elevation amplitude ratio to discuss dead water analogue shows that the bottom permeability is very significant in shallow water waves and intermediate waves. However, for long waves, the bottom permeability is rendered ineffective.

• Water wave trapping in the confined region shows the periodic behaviour for hydrodynamic coefficients. However, a major part of wave dissipation occurs due to bottom permeability, resulting in less waveload.

• The current research is expected to add to our knowledge of the mechanism of a barrier in motion. The analysis of radiated potential amplitude establishes that the radiated potential in free surface mode contains a higher wave flow of energy with increasing frequency. It also shows that the impact of the perforated barrier produces radiated outgoing waves of lower amplitude in both propagating modes.

• The analysis of added mass and damping coefficient establishes that the porosity of the barrier plays a crucial role in energy dissipation. The complex porous-effect parameter of the barrier may give rise to the higher added mass. The higher frequency of the barrier results in a smooth nature for both characteristics.

In a nutshell, the study of water wave interaction with perforated structures enables us to properly understand the influence of such structures that aid in creating a calm zone for the protection of coastal infrastructure. A realistic scenario can be envisaged when both the reflection and radiation of waves are considered.