Large Floating Structures (VLFS).
There are many theories available to deal with a porous bed. For simplicity, we consider the bottom in all works as a porous plate placed over the rigid bottom. The flow inside the plate is considered to be insignificant and we should confine our analysis to porous bottom with fine pores. The water depth is finite and only the boundary condition at the interface between the fluid and the porous sea-bed is used to handle the physical problem as in Martha et al. [66]. The bed permeability has been characterized by a parameterG= ibBρ2ω
µ withbB as the coefficient which has the dimension of length andµ the coefficient of dynamic viscosity [14]. Generally, Gis a complex number as defined in [123] and can be represented byG=GR+ iGI whereGR contributes to resistance effects and GI (imaginary part) contributes to inertial effects. Further, it may be noted that, to ensure propagation of progressive wave at the far field in the open water region,G1 is assumed to be real [63].
1.5 Brief review of previous works
The study of water wave propagation in a two-layer fluid intercepted by a submerged or floating impediment, such as a barrier, breakwater, or any geometric structure, is very essential in coastal and marine engineering. In recent decades, there has been a growing interest in studying this kind of occurrence in order to design suitable porous structures that may be utilised as breakwaters to lessen the effect of waves on shorelines. It is critical to reduce wave forces in order to safeguard different infrastructures as well as shorelines.
This creates a relatively peaceful zone along the coast, allowing operations such as vessel anchoring, loading and unloading of goods, and so on to take place without too much disruption. The goal is to create a calm zone with the least amount of wave effect. The term "tranquilly zone" refers to a region where the sea is relatively calm and the power of waves has diminished significantly. When a train of water waves hits an obstruction on any stratum of the ocean, some of the wave is reflected, while some is transferred across it, and the rest is stuck. The use of suitable porous structures to evaluate the corresponding coefficients will provide a clear situation of how to build a serenity zone that will provide a variety of advantages for the smooth operation of activities on and near shorelines.
Jarlan recommended using perforated breakwaters to lessen wave effect on stiff vertical wall breakwaters installed behind it in 1961 [44]. Sollitt and Cross [95] found and analysed the dissipation of wave energy within a porous medium by using an iterative procedure and the linearized friction termf. Following that, Madsen [61] investigated the reflection and transmission of linear shallow water waves owing to a rectangular porous wave-absorber installed on a level ocean bottom using the linearized version of the governing equations
and the flow resistance formula. Rojanakamthorn et al. [84] employed potential theory to solve a boundary value problem involving porous structures. By studying several forms of rubble-mound breakwaters, which constitute most of the basic porous structures, Sulisz [97] employed boundary element technique to develop numerical solutions to such scattering issues. Dalrymple et al. [25] reported the finding that wavenumbers derived from the dispersion relation inside a porous medium merged in certain situations, and hence the velocity potential within the porous medium could not be represented in terms of eigenfunctions. Yu and Chwang [123] investigated wave propagation through a porous structure by using the virtual work concept developed by Sollitt and Cross [95]. Different approaches for solving the dispersion equation inside a dissipative medium were developed by Mendez and Losada [68] and Chang and Liou [18]. Mendez and Losada [68] utilised a second-order perturbation approach to find higher-order roots, while Chang and Liou [18] used the homotopy perturbation method. The dissipation of wave energy inside porous structures is an important element to be considered while investigating water wave interaction with such structures. For examining regular wave interaction with a caisson-type breakwater with perforated-wall consisting of a perforated front wall and a chamber with an impervious rear wall, Zhu and Zhu [126] developed an impedance analytical technique (IAM). Garrido and Medina [37] proposed a novel semi-empirical model for predicting the reflection coefficient for single and double perforated chamber breakwaters.
Additional research on various forms of porous breakwaters sheds more insight on the subject. In his porous wave-maker theory, Chwang [20] utilised Darcy’s law to investigate waves flowing through a porous structure, and Yu [120] extended it by incorporating the inertia effect in the formulation. The thin-plate model was extensively used by Chwang and Lee [21] and Sahoo et al. [91] to examine waves propagating through various porous structures with diverse topologies. The performance of a breakwater with a perforated front wall, an impervious rear wall, and a rock-filled core was studied by Isaacson et al. [44]. Kaligatla et al. [46] and Tabssum et al. [98] examined water wave interaction with porous structures for waves propagating across sea-beds of varying depths. It was shown that a breakwater made up of numerous porous blocks of varying width may effectively reduce wave energy. Zhu [127] studied the reflection and damping of surface waves propagating through a porous material by using wave-induced refraction-diffraction equations. The problem was solved by using the Sollitt and Cross model in combination with the orthogonality of the depth-dependent functions. Das and Bora [32, 33] built on the work of Zhu [127] to investigate wave reflection and damping by a rectangular porous structure with uniform porosity placed on raised single-step and multi-step bottoms. Liu et al. [58] investigated the hydrodynamic performance of a modified horizontal-plate breakwater consisting of a submerged upper horizontal porous plate and a submerged lower horizontal solid plate by computing the reflection coefficient for various depths,
1.5. Brief review of previous works 21 structure width, and porosity. Sahoo et al. [90] studied the effect of porous structure in the presence of a thin elastic plate.
All these above works focused on wave propagation through or within porous struc- tures mainly in homogeneous fluids only. What was missing and is probably additionally required is to focus on water wave interaction with a porous structure in a stratified ocean which represents a more realistic scenario. Those different strata or layers in ocean form due to differences in temperature and salinity in coastal regions, which results in appre- ciable changes in water density. Thus, in reality, ocean water is non-homogeneous with respect to space and time. In other words, ocean water is formed by different strata, i.e., density layers, with sharp interfaces. Practically, there are three main ‘layers’ in an ocean:
the surface ocean, the deep ocean, and the seafloor sediments (Thorpe [99]). The surface ocean can be thought of as the upper layer. When scientists refer to the surface ocean, the portion of top200meters of the ocean (on average) is referred but the change in depth takes place corresponding to different seasons and latitudes. In general, the surface ocean is much warmer than the deep ocean, and the bottom of the surface ocean is determined by measurements of water temperature and density. The transition into the deep ocean happens when the temperature of water drops and the density increases. The deep ocean is all the seawater that is colder (generally0-3◦ C or32-37.4◦ F), and thus denser. That is why in our consideration, we assume the depth of the lower layer is more than or equal to the upper layer. In general, deep ocean water, which makes up approximately 90%
of the water in the ocean, is homogeneous (it is relatively constant in temperature and salinity from place to place). Because the surface and deep ocean layers are of different densities (due to salt content and temperature), these layers of the ocean do not mix easily. Seafloor sediments are sediments that can be found at the bottom of the seafloor that is still in contact with seawater. These sediments can still be influenced by the wa- ter around them. There are several types of sediments that make up the seafloor, and which are porous. We consider this type of sea-bed as the one for the present problem.
More such practical information can be found in [99]. The surfaces of interfaces between different layers become unstable due to the variation in density gradient, and when some impulse is applied, wave-like disturbances are often generated, which are known as inter- nal interfacial waves. For detail description on interfacial wave and its significance, one can refer to the monograph by Massel [67]. Exchange of momentum and wave energy takes place between the surface and interfacial waves. Subsequently, the effects of wave amplitudes on various marine structures can be studied which leads to full information of interfacial waves impinging upon structures.
Linton and Cadby [56] examined the trapped-mode frequencies for submerged hori- zontal circular cylinders in a two-layer fluid consisting of a finite layer bounded above by a free surface and below by an infinite layer of greater density. Saha and Bora [87, 88]
investigated the existence of trapped modes in a two-layer fluid of finite depth bounded
above by a rigid lid or an ice-sheet. For a fixed geometrical configuration, they numerically computed the values of such wave frequencies for which the values of the truncated deter- minant became approximately zero confirming the existence of trapped modes. Manam and Sahoo [65] used Havelock’s expansion for the potential function to investigate scat- tering and radiation of oblique water waves by a thin porous structure in a two-layer fluid for both finite and infinite depth cases. It revealed that porous structures underwent a substantial resistance by the interfacial waves. Afterward, Behera and Sahoo [8] discussed the effect of porous structures on scattering and trapping of water waves in a two-layer fluid due to a breakwater placed on a permeable sea-bed. The seepage effects by both surface and interfacial waves on the structure were found to be of the same magnitude.
It seems pertinent to discuss a few aspects of use of multi-porosity structures as break- waters. Caisson type breakwaters usually consist of multiple layers (two or three). An enormous breakwater was constructed for the first liquefied natural gas (LNG) receiving terminal along the Pacific Coast of North and South America known as Costa Azul cais- son breakwater. This project successfully demonstrated that very large caissons can be installed in relatively deep water. This is one of the prime examples that has motivated our present work. As an example of another such structure, 40 large concrete caissons were arranged in a specific manner for the Port of Piraeus, West Pier 3 expansion project, just south of Athens, Greece. The individual caisson boxes were built on a large, semi- submersible barge. Now, if we consider one caisson box and its vertical cross-section, then effectively multiple chambers will be formed which also clearly resembles our present work.
Méhauté [79], while discussing wave damping, introduced high wave energy damping due to the progressive wave absorber with multiple layers and analyzed the effect of using mul- tiple damping materials. Experiments were carried out to analyze the effects of two-layer and three-layer porous structures, that is, by using porous blocks with multiple values of porosity and friction factor. Twu et al. [103] considered a stratified porous structure and they were successful in establishing that the multi-porosity structure could be efficiently used as partial shelter to shorelines in order to avoid high wave attack. It was concluded that use of multiple porosities in the breakwater structure added more efficiency to the wave blocking. Lin et al. [54] examined the effect of a horizontal composite porous barrier in linear wave propagation. Venkateswarlu and Karmakar [107] numerically investigated reflection of water waves by multiple porous structures on leeward region. They also considered a stratified porous structure and examined its hydrodynamic performance by computing the reflection coefficient corresponding to various depths, structure width, and porosity [105]. The above works amply justify that use of stratified porous structure has immense contribution in protecting various coastal and offshore facilities by attenuating the wave height.
The sea-bed was thought to be flat and rigid in the wave-structure interaction problems as mentioned above. However, it is important to note that, when gravity waves spread
1.6. Main motivation for the current work 23