1   Introduction

Coastal erosion is a threat to coastal community that can cause various risks, including property damage, loss of land and ecosystem. The construction and use coastal protection structures is necessary to protect coastline and coastal facilities from severe wave action. In order to dissipate the incoming wave energy and thereby aid in the creation of a calm and secure harbourage, porous structures are frequently used in coastal engineering. Several researchers have extensively studied the dissipation of incoming wave energy due to the presence of single-layer porous structure with uniform porosity and friction factor (Sollitt and Cross, 1972; Madsen, 1983; Darymple et al., 1991; Mallayachari and Sundar, 1994; Das and Bora, 2014; Sulisz, 1985) and vertical porous membrane (Koley and Sahoo, 2017; Ashok and Manam, 2022). Experimental studies are also conducted to study the hydrodynamic characteristics of porous structures (Kondo and Toma, 1972; Sollitt and Cross, 1972; Twu and Lin, 1990; Somervell et al., 2017). The design and construction of a stratified structure is found to be an effective solution for dissipating the maximum wave energy and creating a tranquillity zone. In addition, studies are being conducted for the development of new configurations of porous structures in order to achieve maximum wave energy dissipation.

Significant studies have been carried out on the dissipation characteristics of wave interaction with stratified porous structures. Yu and Chwang (1994) examined the wave motion through a porous structure consisting of two layers and noted that a limiting thickness exists for the structure, beyond which hydrodynamic coefficients remain constant with any further increase in thickness. Later, Twu and Chieu (2000) designed an offshore breakwater having minimal wave reflection and transmission using complex eigen‐function method. The numerical and experimental study revealed that a multi-layer breakwater can reduce both wave reflection and transmission at a narrower width compared to a single-layer porous structure. Thereafter, Twu et al. (2002) studied the wave-damping characteristics of a vertically stratified porous breakwater under oblique wave action. Further, Liu et al. (2007) proposed a two-layer horizontally stratified rock-filled core for a perforated break‐water and studied the effect of perforated breakwater on wave reflection and wave force impact on the structure. Recently, Venkateswarlu et al. (2020a) studied the oblique wave transformation due to fully-extended two-layered, three-layered and two-layered submerged horizontally stratified structures using the eigenfunction expansion approach. The study concluded that the increase in porosity of surface layer and moderate friction factor enhances energy damping. The multiple structures are found to be useful if the structure width is high and for a fixed structural width, separating into several structures helps in achieving high wave damping.

Additionally, it is critical to comprehend and understand the effect of seabed topography on the hydrodynamic behaviour of coastal protection structures, as reported by various researchers in the literature. Das and Bora (2014) investigated the wave reflection by a vertical porous structure placed on a stepped seabed as two separate cases with two and multiple steps. The study reported that lower values of friction factor led to oscillation in the reflection coefficient which vanished for higher values of friction factor but for relatively long waves, the values of the angle of incidence did not affect the reflection coefficient whereas short waves resulted in lower reflection coefficient. Afterwards, Hu et al. (2019) conducted an analytical study for oblique scattering of monochromatic small amplitude wave trains by a stationary rigid multi-layered structure of rectangular crosssection for a combined floating and bottom-mounted permeable breakwater. Venkateswarlu and Karmakar (2020b) investigated the significance of seabed characteristics in wave transformation in the presence of a vertically stratified porous structure. The study noted that the wave energy dissipation increases with the increase in porosity of the seaward porous layer. The theoretical results are compared with Twu and Chieu (2000) and found to be in good agreement with the experimental observations. The study suggested that, for better wave blocking, porosity of the leeside porous layer can be kept minimal. The performance in wave reflection of the porous structure placed on a stepped seabed is found better compared to that on a uniform and elevated seabed whereas the structure placed on an elevated seabed showed a significant role in wave blocking. Tabssum et al. (2020) analysed wave interaction with a thick, porous breakwater in a two-layer fluid, which is particularly applicable to continental shelves, having bottom undulation. Later, Venkateswarlu and Karmakar (2020c) examined the wave interaction with multiple porous structures upon elevated seabed in the presence and absence of a leeward wall.

Several researchers performed studies on partial structures which can be combined with primary structures to understand the improvement of wave dissipation properties of composite structures. These partial structures are especially beneficial as they are economic, uses less construction material and has least environmental concerns. Sahoo et al. (2000) investigated the wave interaction with vertical permeable barriers of various configurations including surface-piercing, bottom-touching and fully submerged barrier using eigenfunction expansion and least-squares method. The study concluded that the presence of porous barrier helps in attaining a minimum value of reflection coefficient for a particular value of the porous-effect parameter. Liu and Li (2013) presented a new analytical solution for hydrodynamic coefficients of a surface-piercing porous breakwater. Karmakar and Guedes Soares (2014) investigated the interaction of surface gravity waves with multiple bottom-standing flexible porous barriers. The study suggested that these structures are effective in wave attenuation. Later, Koley et al. (2015) studied wave scattering by a surface-piercing and bottom-standing structure placed at a distance from a rigid wall. Further, Behera and Ghosh (2019) dealt with a surface-piercing flexible porous barrier near a rigid wall in the presence of step-type bottoms and concluded that using suitable combination of wave and structural parameters, this structure can be used as an effective breakwater.

On investigating the studies performed by previous researchers, it is noted that the combination of stratified porous structure with partial porous structure in the presence of elevated step is limited. So, in the present study, wave interaction due to the presence of horizontal and vertical stratified porous structure combined with surface-piercing porous block in stepped sea bottom is examined to understand the effect of the combined stratified structure with surface-piercing porous block on wave attenuation. Numerical modelling is carried out using the eigenfunction expansion method along with orthogonal modecoupling relations. The numerical study on the horizontally and vertically stratified porous structure combined with the porous block is performed and further analysis is performed for the effect of varying porosity, structural width, angle of incidence and length between the porous block and stratified structure. The study performed on the wave interaction with stratified porous structure with stepped bottom is noted to attenuate the wave height in transited region and helps in the dissipation of incoming wave in the large extent. The presence of the stratified structure is observed to largely dissipate the wave energy as compared to single porous structure and the wave trapping phenomenon is also noted due to the stepped bottom resulting in attenuation of wave height. The present study will be very helpful in the wave force reduction on the offshore facility and also useful in the dissipation of wave energy to create the tranquil region.

2   Mathematical formulation for stratified structure with porous block

The present study investigates the oblique wave interaction with stratified porous structure combined with a surface-piercing porous block placed on stepped sea bottom using the small amplitude wave theory. The porous structure is considered to have different layers of porosity in horizontal and vertical directions considering the wave direction. The structure with layers of different porosity is termed as stratified structure. The study is performed on both horizontally and vertically stratified porous structures consisting of two layers of different porosities and friction factors. The surface-piercing porous block is on the seaward side of the stratified porous structure. The horizontally stratified porous structure is placed over the first rigid step whereas the vertically stratified structure is over first and second rigid steps such that each layer is over each step. The thickness of both the porous layers are kept equal throughout the study. The physical problem is analysed in the two-dimensional cartesian coordinate system with x and z-axis in horizontal directions and the y-axis is considered positive downward. The structure is infinitely extended along z-direction with the wave considered to be obliquely propagating in x-direction with an angle θ.

The seabed is assumed to be impervious such that no flow is possible in a perpendicular direction. The porous structure is assumed to be fully-extended and the height is considered equal to the free surface. The fluid domain is divided into nine regions. In the case of horizontally strati‐ fied porous structure (Figure 1), the regions considered include the upstream open sea region I₁^h ≡(0 ≤ x ≤ ∞, 0 ≤ y ≤ h), porous block region I₂^h ≡(− a₁ ≤ x ≤ 0, 0 ≤ y ≤ a), region below porous block I₃^h ≡ (− a₁ ≤ x ≤ 0, a ≤ y ≤ h), region between porous block and stratified porous block I₄^h ≡ (− a₂ ≤ x ≤− a₁, 0 ≤ y ≤ h), stratified structure region I₅^h ≡ (− a₃ ≤ x ≤− a₂, 0 ≤ y ≤ b), bottom stratified structure region I^h₆ ≡ (− a₃ ≤ x ≤− a₂, b ≤ y ≤ h), open sea region above second rigid step I₇^h ≡ (− a₄ ≤ x ≤− a₃, 0 ≤ y ≤ h₃), open sea region above the third rigid step I₈^h ≡ (− a₅ ≤ x ≤− a₄, 0 ≤ y ≤ h₄) and downstream open region I₉^h ≡ (− ∞ ≤ x ≤ − a₅, 0 ≤ y ≤ h₅). In the case of vertically stratified porous structure (Figure 2), the regions include the upstream open sea region I₁^v ≡ (0 ≤ x ≤ ∞, 0 ≤ y ≤ h), porous block region I^v₂ ≡ (− a₁ ≤ x ≤ 0, 0 ≤ y ≤ a), region below porous block I^v₃ ≡ (− a₁ ≤ x ≤ 0, a ≤ y ≤ h), open sea region between porous block and stratified porous block I₄^v ≡ (− a₂ ≤ x ≤ − a₁, 0 ≤ y ≤ h), seaward porous layer over first rigid step I^v₅ ≡ (− a₃ ≤ x ≤− a₂, 0 ≤ y ≤ h₂), leeward porous layer over second rigid step I₆^v ≡ (− a₄ ≤ x ≤− a₃, 0 ≤ y ≤ h₃), open sea region above third rigid step I₇^v ≡ (− a₅ ≤ x ≤ − a₄, 0 ≤ y ≤ h₄), open sea region above the fourth rigid step I₈^v ≡ (− a₆ ≤ x ≤− a₅, 0 ≤ y ≤ h₅) and the downward open sea region I₉^v ≡ (− ∞ ≤ x ≤− a₆, 0 ≤ y ≤ h₆).

Figure  1  Horizontally stratified porous structure with the porous block in the stepped seabed

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Figure  2  Vertically stratified porous structure with porous block in stepped seabed

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It is assumed that fluid is inviscid, incompressible, irrotational and time harmonic with angular frequency ω. The velocity potentials in the respective regions are represented in the form ϕ_j ( x , y, z, t) = Re[ϕ_j ( x , y)e^{i (lz − ωt)}], where l = γ₁₀ sin θ and γ₁₀ is the progressive wave number in open water region. The velocity potentials in the respective regions satisfy the Helmholtz equation given by

The linearized free surface boundary condition for the open water and porous structure region is of the form

where $ K_{j=1, 4, 7, 8, 9}=\frac{\omega^2}{g}, \text { and } K_{j=5, 6}=\frac{\omega^2}{g}\left(S_j+\mathrm{i} f_j\right)$ for free surface and vertically stratified porous structure region, S_j and f_j are reactance and frictional coefficients of porous regions of breakwater and g is the acceleration due to gravity. The bottom boundary condition due to the presence of impermeable sea-bed is given by

The wave propagating due to the presence of the porous structure suggest the continuity of fluid pressure and velocity across the seaward and leeward structural interfaces. The linearized resistance f_j and reactance coefficients S_j (Sollitt and Cross, 1972) offered by each of the porous layer is determined on solving the relation given by

where, C_m is coefficient of added mass considered to be very minimal/zero (Sollitt and Cross, 1972), thus S_j = 1 is kept fixed throughout the study. Λ_p is the intrinsic permeability, q the instantaneous Eulerian velocity vector, ν the kinematic viscosity, V the volume, C_f the turbulent resistant coefficient and T the wave period. The continuity of the velocity and pressure is satisfied for both surface piercing porous block and stratified porous structures. In the farfield region, the radiation conditions in the presence of porous structure is given by

where, I₁₀, R₁₀ and T₉₀ are the complex amplitude of the incident, reflected and transmitted wave energies respectively. In the present study, the incident wave I₁₀ is considered to be unity. The continuity of pressure and velocity due to the presence of surface piercing porous block at the edge x = 0, − a₁ and j = 1, 4 are given by

The wave number in surface-piercing porous block region satisfies the dispersion relation for finite water depth given by

where,

In Eq. 7(a), γ_2n, n = 0, 1, 2, ··· are the wave number for the surface-piercing porous block region which has infinite number of complex roots with the associated eigen value. The real part of the complex wave number, specifies the spatial periodicity (Sollitt and Cross, 1972) whereas the imaginary part specifies the decay rate. The dispersion relation due to the presence of surface-piercing block in Eq. 7(a) is solved using the perturbation method as discussed in Mendez and Losada (2004). The contour plot to determine the location of complex roots of the dispersion relation is presented in Figure 3 considering S₂ = 1.0, f₂ = 0.5, a/h = 0.20 and ε₂ = 0.25. The contour plots are used to find the initial guess for the determination of the roots of the dispersion relations.

Figure  3  Contour plots of roots of dispersion relation of surface-piercing porous block considering S₂ = 1.0, f₂ = 0.5, a/h = 0.20 and ε₂ = 0.25

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2.1   Horizontally stratified porous structure

The horizontally stratified porous structure is assumed to be placed at − a₃ < x < − a₂ on the leeward side of porous block over a rigid step and the water depth 0 < y < h₂. So, the continuity of pressure and velocity due to the presence of two layered porous structure at the edge x = − a₂, − a₃ and j = 4, 7 are given by

In addition, there exists a flow within the surface and bottom porous layers in the vertical direction (Liu et al., 2007) for − a₃ < x < − a₂ and y = b is given by

In the case of change in the bottom topography, the flow near rigid step at x = − a₂, − a₃, − a₄, − a₅ satisfies the zeroflow condition given by

The wave number in upstream/downstream free-water region γ_j0 for j = 1, 4, 7, 8, 9 and porous structure region γ₅₀ satisfies the dispersion relation for finite water depth is given by

where, h_j = h for j = 1, 4, h_j = h₃ for j = 7, h_j = h₄ for j = 8 and h_j = h₅ for j = 9 respectively with

The dispersion relation of open water regions has one real root and infinite imaginary roots which is solved using Newton-Raphson method while the dispersion relation for horizontally stratified porous structure is having infinite number of complex roots which is solved using perturbation method (Mendez and Losada, 2004).

2.2   Vertically stratified porous structure

The vertically stratified porous structure is assumed to be placed at − a₄ ≤ x ≤− a₂ over two rigid steps with water depths 0 ≤ y ≤ a₂ and 0 < y < h₃. So, the continuity of pressure and velocity due to the presence of two layered porous structure at the edges x = − a₂, − a₃, − a₄ are given by

In the case of vertically stratified porous structure, zero flow condition exists at the interface of rigid steps x = − a₂, − a₃, − a₄, − a₅, − a₆ given by

The wave numbers in the upstream/downstream free-water regions are same as Eq. (11a) and the vertically stratified porous structure region satisfy the dispersion relation given by

where h_j = h₂, h₃ for j = 5, 6 respectively. The dispersion relation for the vertically stratified porous structure region is solved using perturbation method (Mendez and Losada, 2004). The contour plot to determine the location of complex roots of the dispersion relation is presented in Figure 4.

Figure  4  Contour plots of roots of dispersion relation of vertically S₅ = 1.0, f₅ = 0.25 and h_s/h = 0.50

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3   Method of solution for stratified structure with porous block

The wave interaction with the stratified porous structures combined with porous block in the presence of elevated sea-bed is examined using the eigenfunction expansion method. In the present section, the solution approach for both horizontally and vertically stratified porous structure is presented in detail. The velocity potential in the incident open water region for both the horizontally and vertically stratified structure over stepped seabed is given by

where, I₁₀ is the amplitude of the incident wave, R_1n for n = 1, 2, 3, ··· are the unknown constants to be determined. The vertical eigenfunction for incident open water region is given by

The velocity potential due to the presence of surface piercing porous block and leeward open water region are given by

where A_2n, B_2n, A₄₀, B₄₀, A_4n, B_4n for n = 0, 1, 2, … are the unknown constants to be determined, w₁ = a₁ is thickness of the porous block. The vertical eigenfunctions for the porous block and leeward open water regions are given by

The eigenfunctions f_jn(y) for j = 2, 3 satisfy the orthogonal mode-coupling relation of the form

Using the continuity of pressure and velocity along with orthogonal mode-coupling relation at the interface x = 0, − a₁, the equation is given by

3.1   Horizontally stratified porous structure

The velocity potentials due the presence of horizontally stratified porous structure and leeward open water regions are given by

where T₉₀ is the transmitted wave amplitude, A_jn, B_jn and T_9n for n = 0, 1, 2, … and j = 5, 6, 7, 8 are the unknown constants to be determined, w₂ = (a₃ − a₂) is thickness of the porous structure. The vertical eigenfunctions for the porous structure and leeward open water regions are given by

where, h' = h₃, h₄, h₅ for j = 7, 8, 9 respectively. The eigen‐functions f_jn(y) for j = 1, 4, 7, 8, 9 satisfy the orthogonal mode-coupling relation of the form

where $ C_{j n}^{\prime}=\left\{\frac{2 \gamma_{j n} h_j+\sinh 2 \gamma_{j n} h_j}{4 \gamma_{j n} \cosh ^2 \gamma_{j n} h_j}\right\}$, for j = 1, 4, 7, 8, 9, n = 0 respectively with $ \left.C_{j n}^{\prime}\right|_{j=1, 4, 7, 8, 9}$ for n = 1, 2, 3, … are obtained by substituting γ_jn = iγ_jn in the case of open water region. Using the continuity of pressure and velocity along with orthogonal mode-coupling relation at the interface x = − a₂, − a₃, the equation is given by

Next, the continuity of pressure and velocity at the edge x =− a₄ along with orthogonal mode-coupling relation is utilized to obtain the equation given by

Further, the continuity of pressure and velocity at the edge x = − a₅ along with orthogonal mode-coupling relation is utilized to obtain the equation given by

The infinite sums presented in the Eqs. 24(a‒b), 25(a‒b) and 26(a‒b) obtained from the orthogonal mode-coupling relation are truncated upto finite M terms to obtain a linear system of 12(M + 1) algebraic equations for the determination of 12(M + 1) unknowns. The wave reflection and transmission coefficient due to the presence of porous structure is given by

Due the existence of porous structure along with stepped seabed the energy dissipation in the wave propagation is represented as

where $ \chi=\left\{\frac{k_{j 0} \tanh \gamma_{j 0} h_j}{k_{10} \tanh \gamma_{10} h}\right\}\left\{\frac{\cosh ^2 \gamma_{10} h}{\cosh ^2 \gamma_{j 0} h_j}\right\}\left\{\frac{2 \gamma_{j 0} h_j+\sinh 2 \gamma_{j 0} h_j}{2 \gamma_{10} h+\sinh 2 \gamma_{10} h}\right\}$

In the next section, the wave attenuation due to the presence of vertically stratified porous structure is examined.

3.2   Vertically stratified porous structure

The velocity potentials in the porous structure and leeward open water regions for the vertically stratified porous structure are given by

where T₉₀ is the transmitted wave amplitude, A_jn, B_jn and T_9n for j = 5, 6, 7, 8, n = 0, 1, 2, … and are the unknown constants to be determined, w₂ = (a₄ − a₂) is thickness of the porous structure. The vertical eigenfunctions for the porous structure and leeward open water regions are given by

where, h_j = h₄ for j = 7, h_j = h₅ for j = 8 and h_j = h₆ for j = 9 respectively. The eigenfunctions f_jn(y), j = 1, 4, 7, 8, 9 satisfy the orthogonality relation of the form

where $ \left.C_n^{\prime}\right|_{j=1, 4, 7, 8, 9}=\left\{\frac{2 \gamma_{j n} h_j+\sinh 2 \gamma_{j n} h_j}{4 \gamma_{j n} \cosh ^2 \gamma_{j n} h_j}\right\}$, n = 0, and $ \left.C_n^{\prime \prime}\right|_{j=5, 6}= \left\{\frac{2 \gamma_{j n} h+\sinh 2 \gamma_{j n} h}{4 \gamma_{j n} \cosh ^2 \gamma_{j n} h}\right\}$, n = 0, 1, 2, … with $ \left.C_n^{\prime}\right|_{j=1, 4, 7, 8, 9}$ for n = 1, 2, 3, … are obtained by substituting γ_jn = iγ_jn in the case of open water region.

Using the orthogonal mode-coupling relation at the interface x = − a₂, 0 ≤ y ≤ h₂ we have

Also, at the interface between the porous layers x = − a₃, 0 ≤ y ≤ h₂, the continuity of pressure and the pressure and velocity along with orthogonal relation gives

The continuity of pressure and velocity at the edge x = − a₄ along with orthogonal mode-coupling relation is utilized to obtain the equation given by

Next, the continuity of pressure and velocity at the edge x = − a₅ along with orthogonal mode-coupling relation is utilized to obtain the equation given by

Further, the continuity of pressure and velocity at the edge x = − a₆ along with orthogonal mode-coupling relation is utilized to obtain the equation given by

The infinite sums presented in the Eqs. 31(a‒b)‒35(a‒b) obtained from the orthogonal mode-coupling in open water regions are truncated upto finite M terms to obtain a linear system of 14(M + 1) algebraic equations for the determination of 14(M + 1) unknowns. The wave reflection, transmission and dissipation coefficients due to the presence of vertically stratified structure are same as in Eq. 27(a‒b).

3.3   Wave force on the front face of porous block and stratified structure

The wave force impact acting on the front face of porous block, K_fb1 and that of stratified structure, K_fs1 are given by

and I₁₀ is the amplitude of the incident wave potential considered to be unity. In the case of surface piercing porous block, F_fb1 is given by

In the case of horizontally stratified porous structure, F_fs1 for j = 5, 6 is given by

In the case of vertically stratified porous structure, F_fs1 is given by

3.4   Surface elevation

The free surface gravity wave elevation in the incident and transmitted wave regions are obtained from the relations

which can be expressed in the form as

where, − a_j = − a₅, h_j = h₅ for horizontally stratified structure and − a_j= − a₆, h_j = h₆ in the case of vertically stratified porous structure. The numerical investigation performed using the eigenfunction expansion method is for the regular geometry of the stratified porous structure and elevated sea-bed. In the case of irregular geometry, the numerical approach using the eigenfunction expansion method can be coupled with Boundary Element Method (BEM) to perform the numerical investigation.

4   Numerical results and discussion for stratified structure with porous block

The numerical investigation is performed to examine the wave interaction due to stratified porous structure combined with surface-piercing porous block in changing bottom topography considering various values of porosity ε, linearized friction factor f, angle of incidence θ, finite spacing between the structures and porous block L. The wave reflection K_r, transmission coefficient K_t, energy dissipation K_d, wave force impact on the front face of the porous block K_fb1, wave force impact on the front face of the stratified structure K_fs1 and the surface deflection in the incident and transmitted region ζ_j ( x ) is plotted to understand the behaviour of porous structure with change in bottom topography for wave energy dissipation. The convergence study in K_r and K_t due to the increasing number of evanescent wave modes M is performed for horizontally and vertically stratified porous structure combined with a surface-piercing porous block placed on stepped sea bottom and the numerical results obtained are tabulated in Table 1. The numerical result is noted to converge with the increase in the number of evanescent wave modes M ≥ 15 as tabulated in Table 1. In the present study the evanescent wave mode is truncated for M = 15 and the numerical results are evaluated. The parameters that are kept constant are ρ = 1 000 kg/m³, g = 9.81 m/s² and S_j = 1 throughout the computation.

4.1   Validation of the numerical model

The numerical modelling of single layered and multilayered porous structures for different configurations is performed by Dattatri et al. (1978), Dalrymple et al. (1991), Zhu and Chwang (2001) and Liu and Li (2013). The study conducted on a single layer porous structure is validated with results from the literature in order to assess the validity of the present numerical model based on the eigenfunction expansion method. Dalrymple et al. (1991) examined the reflection and transmission characteristics for oblique wave incidence on a vertical porous structure using plane wave approximation.

In Figure 5 the comparative study between the present numerical model and the results obtained for single porous structure on wave reflection coefficient for varying angle of incidence is performed and a good agreement with the result obtained by Dalrymple et al. (1991) is noted. The study reveals that, with the increase in the linearized friction factor, the minimum in the reflection coefficient is obtained for 65° < θ < 75° Further, Liu and Li (2013) developed an analytical solution for wave reflection and transmission by a surface-piercing porous breakwater without using the complex dispersion relation. The results obtained for the reflection and transmission coefficient (Figure 6) based on the analytical solution of single layer porous structure as in Liu and Li (2013) is compared with that result obtained using the present numerical approach. The study shows a considerable agreement between the results using the present numerical approach and by Liu and Li (2013) for both reflection and transmission coefficients. Thereafter, the numerical investigation is extended for both horizontally and vertically stratified porous structure combined with porous block in varying seabed.

Figure  5  Comparative study for the reflection coefficient in the case of single porous structure and Dalrymple et al. (1991) considering B/h = 1, ω²h/g = 0.2012, S = 1 and ε = 0.45

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Figure  6  Comparative study for K_r and K_t using the present analytical approach for single porous structure and Liu and Li (2013) considering ε = 0.45, S = 1 and f = 1

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4.2   Horizontally stratified porous structure with surface-piercing porous block

The wave transformation due to horizontally stratified porous structure combined with porous block in changing bottom topography is analyzed on studying the wave reflection coefficient K_r, transmission coefficient K_t, dissipation coefficient K_d, surface displacement ζ_j ( x ), wave force acting on the front face of porous block K_fb1 and on the stratified structure K_fs1.

4.2.1   Reflection, transmission and dissipation coefficient

The wave reflection, transmission and dissipation coefficients are analysed for the change in the porosity, change in structural width, change in dimensionless wave number, change in the angle of incidence.

4.2.1.1   Effect of multiple porosities

In order to analyze the effect of porosity on hydrodynamic coefficients, the behaviour of combinations of different porosity for top and bottom porous layer of stratified structure is studied in Figure 7(a‒b) for varying dimensionless width of structure. The stratified porous structure with uniform porosity of 50% is observed to have more wave reflection and intermediate transmission compared to the other combinations. The wave reflection shows intermediate characteristics, and transmission coefficient is least for combination of ε₅ = 0.7 and ε₆ = 0.3. As a result, wave dissipation is higher for the porosity combination ε₅ = 0.7 and ε₆ = 0.3. A mono resonating behaviour is observed for K_r and maximum peak is observed when width of porous structure is same as the depth of incident open water region. The maximum wave reflection may be due to constructive inference between incident and reflected waves by the porous structure. Thus for ε₅ = 0.7 and ε₆ = 0.3, K_r for uniform porosity is 33.33% more than that of combination of ε₅ = 0.9 and ε₆ = 0.1. Most of the surface concentrated waves may be either attenuated or reflected by the surface-piercing porous block and hence only the remaining incoming wave energy are attenuated by the stratified structure.

Figure  7  Variation of K_r, K_t and K_d versus w₂/h for different combinations of porosities considering γ₁₀h = 1.5, h_s/h = 0.10, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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In Figure 7(a), it can be noted that K_r is decreasing as the porosity of surface layer is increasing which may be caused due to the easy penetration of waves through the pores of surface layer of stratified structure. This is evident from K_t in Figure 7(a) that with the maximum wave transmission is observed for the combination of highest porosity at the surface layer. Further, about 95% wave energy dissipation is achieved when width of stratified structure is more than 1.5 times the water depth as depicted in Figure 7(b).

4.2.1.2   Effect of structural width of porous block and stratified structure

Figure 8 illustrates the hydrodynamic coefficients versus θ for different structural width of the porous block varying within 0.5 < w₁/h < 3.0. The increase in the structural width w₁/h shows an increase in wave reflection but a decrease in wave transmission. For direct wave attack, K_r for w₁/h = 3.0 is observed to be twice as compared to w₁/h = 0.5. Also, a small mono resonating behaviour in K_r is prominent at θ = 22° as width of porous block increases. This can be due to the change in phase of the incident and reflected wave due to the increased width of the porous structure. For higher angle of incidence, a steep increase in wave reflection is observed.

Figure  8  Variation of K_r, K_t and K_d versus θ for different values of w₁/h considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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As depicted in Figure 8(b), wave energy dissipation decreases with increase in width of porous block, although larger variation is not observed for oblique waves with angle of incidence greater than θ = 45°. At θ = 15°, K_d shows a decrease of 65.28% from w₁/h = 0.5 to w₁/h = 3.0. The decrease in wave energy dissipation is caused as most of the incoming waves are being reflected by the porous block. A local minima is observed in K_d for w₁/h = 3.0 due to the sudden increase in wave reflection. A steep decrease in K_d is observed for higher angle of incidence which is an opposite trend to that of Kr. This may be due to the less effective wave trapping as most of this incoming wave energy is being reflected by the porous structure.

The hydrodynamic behaviour of structure with respect to different structural width of stratified porous structure as well is analysed for varying angle of incidence. In Figure 9(a) the variation in K_r is observed, which increases with more oblique wave angle. Also wave reflection is more when width of structure is half of the open water depth. In all other cases, no significant variation in K_r is observed. Wave transmission characteristics for w₂/h = 0.5 also suggests that this structural width is not suitable for stratified structure as K_t is more. This is due to the availability of shorter path through the stratified porous structure and hence lesser pore spaces for the wave energy to be dissipated. For θ = 0°, wave transmission decreases by around 82% when w₂/h is increased from 0.5 to 3.0. Wave damping efficiency for most cases, except for w₂/h = 0.5 is greater than 90% as noted in Figure 9(b). Since, for all other cases, wave damping efficiency is almost the same, it is economical to construct stratified porous structure with intermediate structural width and to decrease wave transmission for this width of stratified structure, it is better to increase the width of surface-piercing porous block. Thus, cost can be considerably reduced due to the combination of a fully extended structure with a partial porous structure.

Figure  9  Variation of K_r, K_t and K_d versus θ for different values of w₂/h considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.2.1.3   Effect of dimensionless wavenumber

To examine the behaviour of waves in the presence of horizontal stratified structure combined with porous block in changing bottom topography the dimensionless wave number ranging from 0.75 to 1.75 is varied. In Figure 10, the wave reflection, transmission and dissipation coefficients are analysed for varying length between stratified structure and porous block with the change in the non-dimensional wave number. Due to high wave oscillations in the confined region, resonating behaviour is observed for the hydrodynamic coefficients which allow us to determine the optimum length to be provided between stratified structure and porous block from the resonating troughs. The minimum value in wave reflection as observed in Figure 10(a) corresponds to the destructive interference of the waves due to the presence of the porous block. It is observed that wave reflection for shorter waves is more compared to longer waves. Further, wave transmission for longer waves is more. This is due to the lesser interaction of longer waves compared to the length of the structure and vice versa for shorter waves. In addition, as represented in Figure 10(b), wave damping efficiency for shorter waves is higher for the structure, although more prominent resonating behaviour is observed for such waves. However, the oscillating pattern diminishes for lesser value of γ₁₀h and more uniform value is achieved due to the formation of the standing waves in the confined region formed by the porous block in seaward side and rigid steps in leeward side with the horizontally stratified structure.

Figure  10  Variation of K_r, K_t and Kdversus L/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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In Figure 11 the effect of dimensionless wavenumber for varying dimensionless width of stratified structure is presented for varying dimensionless width of stratified structure. For shorter waves, wave reflection is more, and transmission is minimum compared to longer waves due to increased wave interaction with the structure as illustrated in Figure 11(a). In addition, as the width of stratified structure is increased, K_r achieves more consistent value and hence any further increase in width of structure has less effect on wave reflection characteristic. The local maxima observed in K_r for 0.5 < w₂/h < 1.5 may be caused due to the constructive interference between incoming and reflected waves. However, this local maximum value shifts towards left with the increase in γ₁₀h, which can be due to the change in phase of incident and reflected waves when the wavelength of incoming waves changes. Also, even for small structural width, wave dissipation behaviour for shorter waves is efficient as presented in Figure 11(b). A local minima is observed in the case of γ₁₀h = 1.75 because of high wave reflection on account of standing wave formation. Hence, this structure is economical for waves with shorter wavelength. In the case of an increased width of stratified structure, longer waves can also be attenuated effectively by better interaction of wave with structure.

Figure  11  Variation of K_r, K_t and K_d versus w₂/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.2.1.4   Effect of angle of incidence

The impact of angle of wave attack on the wave transformation is examined for the stratified porous structure combined with porous block for stepped seabed. The hydrodynamic coefficients are studied varying angle of incidence within 0 ≤ θ ≤ 90° in Figure 12(a‒b) for different combinations of porosity of surface and bottom layers of stratified structure. It can be inferred from Figure 12(a) that, for more oblique waves, θ > 60°, K_r increases considerably, whereas transmission decreases due to increased wave interaction with structure. Further, for higher angle of incidence, wave dissipation property of the structure as shown by Figure 12(b) decreases. This may be caused due to increased reflection, as a result of which only marginal number of waves would be passing through the porous structure and hence lesser wave energy attenuation of incoming waves.

Figure  12  Variation of K_r, K_t and K_d versus θ for different combinations of porosities considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.2.1.5   Effect of length between porous block and stratified structure

The variation of hydrodynamic coefficients with the change in the length between porous block and the stratified structure for varying combination of porosity of surface and bottom porous layer is studied and presented in Figure 13. An oscillating pattern is obtained for all the cases of combinations of porosity. This oscillation diminishes as the porosity of surface layer is increased and that of bottom layer is decreased. This may be due to the formation of standing waves in the confined region between surface-piercing porous block and stratified structure. In addition, the trough is observed for L/h having the value 1.0, 3.0, 4.0 and so on. The L/h values are significant for designing the structure as we can achieve minimal wave reflection. In these cases, even zero wave reflection is observed in Figure 13(a). This may be the case when incident and reflected waves are 180° out of phase and all the waves are absorbed within the confined region. Also, reflection coefficient decreases as the porosity of surface layer is increased and the bottom layer is decreased. This is due to the availability of more pore spaces in surface layer to allow more surface waves to pass through. Resonating peaks are also observed in the case of K_t, such that the value is increasing with increase in porosity of surface layer. However, wave damping efficiency is maximum shown for the combination with ε₅ = 0.7 and ε₆ = 0.3. Resonating crests for K_d as depicted in Figure 13(b) are observed for L/h corresponding to negligible or fairly zero wave reflection due to the absorption of incoming wave energy.

Figure  13  Variation of K_r, K_t and K_d versus L/h for different combinations of porosities considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.2.2   Surface elevation

The surface elevation in the open water regions for various angle of incidence of gravity waves in the range 0 ≤θ ≤ 60° is analysed in Figure 14. The maximum surface elevation is observed in the case of θ = 60° for the incident open water region. However, irrespective of incident wave angle, considerable reduction in surface elevation of transmitted region is achieved due to the presence of the structure.

Figure  14  Surface elevation versus x for different values of θ considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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Most attenuation is achieved in the case of higher angle of incidence due to considerably more interaction of oblique waves with the structure. Phase change in the incoming waves is observed in Figure 14 due to the presence of the structure. Thus, it is clear that a tranquil zone can be created in the leeward side by the combination of stratified structure with surface-piercing porous block.

4.2.3   Wave force on front face of porous block and stratified structure

Figure 15 shows the force acting on front face of both horizontally stratified porous structure and the surface piercing porous block varying dimensionless wavenumber. It is observed that K_fs1 is less when longer waves are incident as noted in Figure 15(a) This is because longer waves have least interaction with structure and hence less reflection.

Figure  15  Variation of K_fs1 and K_fb1 versus L/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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The resonating behaviour gradually diminishes as the wavelength increases and tend to attain a more uniform value. For the same L/h value, when γ₁₀h = 1.75 attains the peak value, K_fs1 is 86% more compared to that of γ₁₀h = 0.75. The same trend is observed for K_fb1 as in Figure 15(b). On varying angle of incidence of wave, the wave force on the front face of the stratified structure and porous block is studied for different dimensionless width of porous block as in Figure 16(a ‒ b). Both K_fs1 and K_fb1 is decreasing with increase in angle of incidence. For w₁/h = 3.0, K_fs1 is maximum for θ < 10° and minimum for θ ≥ 20°. From Figure 16(a), it is clear that, as the structural width of porous block is increased, wave transmission to the confined region between porous block and stratified structure decreases and hence considerably less wave would be incident on the stratified structure. As a result, wave force acting on the stratified structure would be less. Thus, with the presence of surface-piercing partial structure in combination with a fully extended structure, wave force acting on our primary structure can be significantly reduced. As depicted in Figure 16(b), wave force acting on the front face of the porous block increases with increase in width of porous block as most of the incident wave energy is being reflected by the surface-piercing structure.

Figure  16  Variation of K_fs1 and K_fb1 versus θ for different values of w₁/h considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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Further, to study the effect of structural width (Figure 17) on wave transformation, w₁/h is varied from 0.5 to 3.0 against γ₁₀h. There is considerable variation in K_fs1 for different w₁/h and it is observed to have peak value at around γ₁₀ = 1.5 for any structural width, as in Figure 15. The wave force on stratified structure is observed to be decreasing for wider surface-piercing porous blocks. Beyond γ₁₀h = 5, for smaller structural widths, K_fs1 again increases whereas for larger structural widths, K_fs1 becomes constant.

Figure  17  Variation of K_fs1 versus γ₁₀h for different values of w₁/h considering h_s/h = 0.10, a/h = 0.25, L/h = 0.25, w₂/h = 0.50, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.3   Vertically stratified porous structure with surface-piercing porous block

The wave transformation due to the vertically stratified porous structure combined with porous block in changing bottom topography is analyzed on studying the wave reflection coefficient K_r, transmission coefficient K_t, dissipation coefficient K_d, surface displacement ζ_j ( x ), wave force acting on the front face of porous block K_fb1 and on the stratified structure K_fs1.

4.3.1   Reflection, transmission and dissipation coefficient

The wave reflection, transmission and dissipation coefficients are analysed for the change in porosity, change in structural width, change in dimensionless wave number and change in the angle of incidence.

4.3.1.1   Effect of multiple porosities

In Figure 18, the vertically stratified porous structure combined with porous block in the presence of stepped bottom topography is analyzed for hydrodynamic coefficients, varying the dimensionless width of porous block and for different combinations of porosities of seaward and leeward porous layers. A mono resonating behaviour is observed for K_r in Figure 18(a) which becomes more prominent as porosity of seaward layer increases. Minimum K_r is observed for the combination of ε₅ = 0.9 and ε₆ = 0.1 when w₂/h = 0.75. Thus, this structural width can be used while designing the structure to achieve minimum wave reflection. The reason for zero wave reflection may be that the incoming and reflected waves become 180° out of phase. The K_t value is also considerably reduced as porosity of seaward layer increases and that of leeward porous layer decreases. This can be due to more wave trapping while passing through the porous layers. Compared to horizontally stratified structure in combination with porous block, the combination of vertically stratified structure and porous block showed less K_r and K_t, resulting in more wave energy dissipation as given in Figure 18(b). This may be because, most of the surface concentrated waves are being reflected and dissipated by the surface-piercing porous block and the remaining incident on the stratified structure would be reflected and trapped effectively within the porous layers of vertically stratified structure. The effective trapping is possible when the difference between porosity of seaward and leeward porous layer become larger as observed in the combination of ε₅ = 0.9 and ε₆ = 0.1.

Figure  18  Variation of K_r, K_t and K_d versus w₂/h for different combinations of porosities considering γ₁₀h = 1.5, h_s/h = 0.10, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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The hydrodynamic characteristics of vertically stratified porous structure combined with porous block is again studied with various combinations of porosity varying dimensionless wavelength. In Figure 19 it is observed that the reflection coefficient is same for all combinations of porosity except that the sudden drop of K_r shows different minimum value for different combinations. This sudden reduction in K_r may be due to the destructive interference. The critical wavelength for which reflection coefficient is minimal is observed to be near to λ/h = 3.0. The critical wave‐length decreases as the porosity of seaward layer is increased and that of leeward layer is decreased. Further, for dimensionless wavelength beyond the critical wavelength, an increase in K_r is observed. In the case of wave transmission coefficient, a considerable decrease in K_t is observed for ε₅ = 0.75 and ε₆ = 0.15. Thus, this combination of porosity may be suitable while designing the vertically stratified structure to achieve less reflection and transmission simultaneously and thereby more wave energy attenuation.

Figure  19  Variation of K_r and K_t versus λ/h for different combinations of porosities considering h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.3.1.2   Effect of structural width of porous block and stratified structure

To examine the dependence of hydrodynamic properties on structural width of the structure, the wave reflection, transmission, and dissipation coefficients versus λ/h is studied for 0.5 < w₁/h < 2.0. It can be inferred from Figure 20(a) that due to the presence of the structure, wave reflection coefficient first decreases up to a minimum value and then achieves a nearly stable value for λ/h > 4. Also, for wider structures, as observed in the case of horizontally stratified structure, reflection coefficient is more while the transmission coefficient is less. However, the decrease in K_r is more prominent than increase in K_t for less wider structures, which helps in improving the wave dissipation characteristics as observed in Figure 20(b). For λ/h = 2, a decrease in w₁/h from 2.0 to 0.5, increased K_d by four times. A sudden drop in K_d is observed for w₁/h = 2.0 due to high wave reflection and less transmission of incoming wave energy.

Figure  20  Variation of K_r, K_t and K_d versus λ/h for different values of w₁/h considering h_s/h = 0.10, a/h = 0.25, w₂/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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Further, to understand the significance of width of stratified structure in influencing hydrodynamic behaviour, different values of w₂/h in the range 0.5 ≤ w₂/h ≤ 3.0 are varied against dimensionless wavenumber. It can be inferred from Figure 21(a) that, for shorter waves, K_r is nearly the same irrespective of the width of stratified structure. However, for further increase in λ/h, K_r sharply decreases to a critical value and then increases. The local minima shift towards right with increase in width of stratified structure, may be due to the phase shift when waves encounter more wider structure. In addition, the wave transmission decreases with increase in structural width. For shorter waves, K_t is less but K_d remains same for any structural width. As observed in the case of horizontally stratified structure, longer waves require wider sections for effective wave attenuation. Thus, for these longer waves, wave dissipation of upto 97% can be achieved (Figure 21(b)) by the design of wider structures.

Figure  21  Variation of K_r, K_t and K_d versus λ/h for different values of w₂/h considering h_s/h = 0.10, a/h = 0.25, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.3.1.3   Effect of angle of incidence

The impact of angle of incidence in wave transformation by vertically stratified porous structure combined with porous block is studied for 0 < θ < 90° varying porosity of seaward and leeward porous layer. In the case of higher angle of incidence, the K_t seems to be considerably less while K_r steeply increases as observed in Figure 22(a). In comparison to horizontally stratified structure, although wave reflection is nearly same transmission can be considerably reduced by the construction of vertically stratified structure of same structural width in combination with the partial porous structure. Wave damping efficiency for any angle of attack less than 60° has a constant value, beyond which K_d sharply decreases (Figure 22(b)).

Figure  22  Variation of K_r, K_t and K_d versus θ for different combinations of porosities considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.3.1.4   Effect of dimensionless wave number

In Figure 23, the hydrodynamic coefficients for the vertically stratified structure combined with porous block in the presence of stepped bottom is analysed for 0.75 < γ₁₀h < 1.75 varying dimensionless width of stratified structure, w₂/h. Slight resonating behaviour is observed for Kr with the minimum value being shifted towards right as the wavenumber decreases due to phase shift and destructive interference. The local maxima correspond to the situation when there is constructive interference between incoming and reflected waves. K_r further increases to achieve a fairly stable value, which increases with decrease in wavelength.

Figure  23  Variation of K_r, K_t and K_d versus w₂/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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Wave transmission is on a slightly higher side for longer waves. Compared to the hydrodynamic behaviour of horizontally stratified structure in combination with the porous block, wave reflection and transmission for vertically stratified structure is significantly reduced as observed in previous sections. This is because of effective wave trapping by the vertical layers of stratified structure. For Kd, as shown in Figure 23(b), variation is observed particularly for longer waves only, in the range of 0.5 < γ₁₀h < 3. In contrast to horizontally stratified structure, wave damping was also efficient even for less wider structures.

4.3.2   Surface elevation

In Figure 24, the surface elevation in the incident and transmitted regions for various angle of incidence is observed. The maximum surface elevation is observed in the case of θ = 60° in the incident open sea region, although there is no considerable variation for different angle of incidence. The reduction in surface elevation by vertically stratified structure is more efficient compared to that of horizontally stratified structure in combination with surface-piercing porous block. There is nearly 60% more decrease in surface elevation for vertically stratified structure. Thus, it is more advisable to construct a vertically stratified porous structure combined with porous block for more tranquil zone in the leeward side.

Figure  24  Surface elevation versus x for different values of θ considering γ₁₀h = 1.5, h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, L/h = 0.25, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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4.3.3   Wave force on front face of porous block and stratified structure

The wave force on the front face of the stratified structure and porous block is observed to show a resonating behaviour, for varying length between porous block and stratified structure and is plotted for different values of γ₁₀h as shown in Figure 25. The wave force due to shorter wave is although more on porous block, it is significantly less when acting on the stratified structure. As observed in Figure 25(b), harmonic crests and troughs tend to be increasing with increase in wavenumber due to more interaction of wave with porous block. Optimum length between porous block and stratified structure so as to have minimum wave force can be determined from the plot while designing the structure. In contrary to the wave force acting on horizontally stratified structure, vertically stratified structure combined with porous block is subjected to less impact of waves. This can be due to less wave reflection and more wave trapping within the structure.

Figure  25  Variation of K_fs1 and K_fb1 versus L/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = w₂/h = 0.50, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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In order to study the wave force on the front face of vertically stratified structure K_fs1, different values of w₁/h is considered for varying λ/h. Wave force seems to be increasing with wavelength to attain a peak value and then decreases with further increase in wavelength. This may be due to more reflection that is taking place from the structure for that particular wavelength as a result of constructive interference. Also, it is observed in Figure 26 that wave force decreases with increase in structural width of porous block. This is because, a major part of incident wave energy would be reflected by the wider surface-piercing porous block creating less impact on the stratified structure. However, there are optimum widths of porous block for which wave force can be kept minimal.

Figure  26  Variation of K_fs1 versus λ/h for different values of w₁/h considering h_s/h = 0.10, a/h = 0.25, w₂/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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In Figure 27(a‒b), the wave forces K_fs1 and K_fb1 versus w₁/h is analysed for values of γ₁₀h. The harmonic crests and troughs are observed for both K_fs1 and K_fb1 in such a way that harmonic peak is noted increasing with increase in width of porous block. This is more visible in the case of shorter waves due to increased wave interaction with structure and hence more constructive interference takes place between the incident and reflected waves. The values of w₁/h for which wave force is minimum correspond to the optimum width of porous block. As observed in the previous case, K_fb1 (Figure 27(b)) is considerably more for longer waves.

Figure  27  Variation of K_fs1 and K_fb1 versus w₁/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₂/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.4, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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Further to understand the effect of wave force due to varying width of stratified structure, K_fs1 and K_fb1 are plotted for varying w₂/h for different values of γ₁₀h in Figure 28. For smaller structural width, mono resonating behaviour is observed in K_fs1 such that shorter waves tend to show peak value at lesser widths of stratified structure due to constructive interference between incoming and reflected waves. The same is observed in the case of K_fb1. However, wave forces tend to attain a stable value for wider structures.

Figure  28  Variation of K_fs1 and K_fb1 versus w₂/h for different values of γ₁₀h considering h_s/h = 0.10, a/h = 0.25, w₁/h = 0.50, L/h = 0.25, θ = 20°, S₂ = S₅ = S₆ = 1, ε₂ = 0.2, ε₅ = 0.7, ε₆ = 0.3, f₂ = 0.6, f₅ = 0.8 and f₆ = 0.5

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5   Conclusions

The wave transformation due to horizontally and vertically stratified porous structures combined with surface piercing porous block in changing bottom topography is analyzed based on eigenfunction expansion method. The conclusions drawn from the present study are as follows:

• In the case of horizontally stratified structure, least transmission can be achieved for combination of ε₅ = 0.7 and ε₆ = 0.3 along with 20% porosity for surface-piercing block.

• Wave energy dissipation decreases with increase in width of porous block. K_d shows a decrease of 65.28% from w₁/h = 0.5 to w₁/h = 3.0 in case of horizontally stratified structure combined with porous block.

• In the case of direct angle of incidence, wave transmission decreases by around 82% when width of horizontally stratified porous structure is increased from 0.5 to 3.0 times the water depth.

• Wave reflection for shorter waves is more compared to longer waves. However, wave transmission is least and wave attenuation can be easily achieved due to the presence of the porous block. For longer waves, wave dissipation of upto 97% can be achieved by designing wider structures.

• The wave force acting on stratified structure can be decreased if the structure is combined with wider surface-piercing porous blocks.

• In the case of vertically stratified porous structure, minimum wave reflection can be achieved for the combination of ε₅ = 0.9 and ε₆ = 0.1 when width of the stratified structure is 0.75 times the water depth.

• The presence of stratified porous structure combined with porous block decreases surface elevation to a large extent.

• Vertically stratified structure combined with the surface-piercing porous block seems to be more efficient in terms of less wave reflection and transmission and thereby high wave energy dissipation compared to vertically stratified structure.