Conclusion

uated for higher porous-effect parameter values of the sea-bed. Meanwhile, transmission decreases gradually with the increase of bed porous-effect parameter in free surface mode.

Although negligible difference is noticed for reflection and transmission coefficients for IM, but for |R₂|, a difference is observed between higher and lower bed porous-effect parameter values. The structural width presents a major contribution in the variation of reflection and transmission but after some width, the structural width does not make impact anymore as observed earlier. Therefore, a porous sea-bed dissipates the incident wave energy, and increasing values of porosity realizes higher dissipation, which may be a reason for this variation. Furthermore, the physical properties of the structure also produce a significant contribution regarding both reflection and transmission phenomena.

Furthermore, with respect to the waveload on the wall, visible difference is observed in the trapping model with the incorporation of the sea-bed porosity in which case a lower waveloadK_w on the wall occurs, contrary to the case for an impermeable sea-bed for which a higher waveload is observed on the wall. It is worthwhile mentioning that a major portion of the wave energy gets dissipated through the porous sea-bed which may have resulted in the lower impact of waveload on the rigid sea-wall compared to the case when the sea-bed is impermeable.

3.6 Conclusion

This work studies oblique water wave interaction by a composite porous structure, with different block-wise porosity, in a two-layer fluid flowing over a porous sea-bed followed by an elevated rigid bottom on which the structure is placed and where the structure pierces the linear interface. Moreover, a similar geometrical configuration with a perforated front wall on the porous structure is also considered to examine the wave interaction. Linear water wave theory and eigenfunction expansion are utilized to find the solution of the corresponding scattering and trapping problems. The region is divided into a number of sub-regions, and by using the given equations and matching conditions, a system of linear algebraic equations is obtained and solved which in turn give the potentials and the reflection and transmission coefficients. Upon comparing a present result with an available result, an excellent agreement between them can be noticed which validates the usefulness of the present model which confirms that the investigation could be carried forward. Thereafter, a number of numerical results are obtained and analyzed. The incident, reflected and transmitted waves for the scattering problem and elevations in the free surface and interface exist due to wave interaction with the porous structure. Due to the mutual interaction of the waves in both propagating modes, we observe a greater waveload on the porous structure which reduces sharply at the end of the structure.

For the trapping problem, less waveload on the vertical rigid wall is also observed due to the dissipation of the breakwater. This study also leads us to find the optimum width with a suitable structural configuration, which produces a more efficient porous structure possessing both reflective and dissipative characteristics. Consideration of moderate val- ues of the angle of incidence allows one to achieve minimum wave reflection through the utilization of a porous composite structure as a breakwater. By an appropriate selection of structural height, width, porosity and porous impedance parameter, one can find (i) the maximum wave reflection by the structure, (ii) the maximum dissipation of energy by the porous body, and (iii) the minimum value of the waveload on the wall. Also, maximum wave reflection by the interface-piercing structures can be achieved with the consideration of an appropriate size of the elevated bottom upon which the structure is placed. We also find that a composite porous structure is more efficient in producing higher reflection and lower waveload compared to a simple kind of structure. In comparison, for the structure with a perforated front wall, our observations are as follows: (i) the percolative property of the front wall makes the breakwater more effective in producing higher reflection in both propagating modes, and (ii) better dissipation of the energy reflects lower waveload acting on the rigid wall.

Further, due to the consideration of the sea-bed to be porous, it is observed that the porosity of the sea-bed has a reasonable impact on reflection and transmission character- istics and the waveload due to oblique wave interaction with the porous structure under consideration. The comparison between the porous and impermeable sea-bed shows that a significant effect of porosity on the waveload as well as in the reflection and transmis- sion phenomena is visible due to the bed characteristics. Therefore, study of water wave interaction with such structures presents a clear understanding of the effects of the porous structures and the porous sea-bed that together help in creating a tranquillity zone. It also demonstrates that there is some kind of relation among the optimum wave reflection by the porous structures, the optimum waveload on the rigid wall and the maximum wave load on the porous structure and vice-versa. It can be clearly observed that use of a com- posite porous structure with suitable geometrical configuration along with appropriate characteristics of the sea-bed is a more viable option in creating a tranquillity zone to protect the coastal facilities appropriately.

CHAPTER 4

Interaction of oblique water waves with a single chamber caisson type breakwater for a two-layer fluid flow over an elastic bottom

4.0.1 Mathematical formulation

Figure 4.1: Definition sketch of wave scattering due to interface piercing structure An interface-piercing porous breakwater is considered in a two-layer fluid flowing over an elastic sea bottom. Linear water wave theory and small amplitude bottom deflection in finite ocean water depth are taken into consideration to discuss the oblique surface wave interaction with the breakwater. By assuming large length of the bottom, the thin elastic plate theory can be employed for the elastic bottom, as being considered extensively for problems on wave-structure/ice-sheet interaction problems. The elastic sea bottom is considered as only a boundary in this analysis, and the fluid motion beneath it is not investigated. The flexible nature of the elastic plate renders the condition at the bottom to a fifth order condition which is different from the usual Neumann condition applicable for a rigid bottom. The motion is assumed to be irrotational and simple harmonic, and the

fluid to be inviscid and incompressible consisting of two layers under the action of gravity.

The interface between the layers is taken to be linear. A right-handed coordinate system is employed in such a way that z = 0represents the interface of the layers with the z-axis pointing upwards. Further, the mean free surface of the upper fluid layer of density ρ₁ is located atz =d whereas the lower fluid layer of densityρ₂ is bounded below by an elastic bottom−∞< x <∞with the mean depthz =−h. Based on the previous studies, it can be inferred that the wave force acting on the shoreline is a recurring phenomenon, and placing a porous structure is one of the ways to reduce the wave force on the shoreline.

The present study elaborates the importance of the porous structure standing upon a rigid foundation in protecting the shoreline. The porous structure is used as a caisson type breakwater above a rock bottom foundation located in 0 ≤ x ≤ B. We consider an incident wave obliquely incident at an inclination θ to the x-axis on the structure.

Subsequently, Φ_j, j = 1,2,3, which denote the velocity potentials in Regions 1, 2 and 3, respectively, can be written as Φ_j(x, y, z, t) = Re[φ_j(x, z)e^i(kyy−ωt)]. The schematic diagram of the problem is depicted in Figure 4.1. The variation of each potential Φ_j (j = 1,2,3) in the y-direction is considered to be identical in order that all Φ_j along the vertical boundaries can be matched according to Snell’s Law. A rock foundation is considered on which the caisson type breakwater is placed. Assuming this, we consider an impermeable bottom for porous Region 2 and consequently, elastic bottom effect exists for Regions 1 and 3 only. Furthermore, the effect of the bottom underneath the rigid block is ignored. In order that this specific arrangement is physically viable and can be used for practical problems, justification is provided in Appendix E.

In order that the problem can be clearly understood layer-wise, φ_j(x, z) are split in the following manner:

φ_j =







ψ_j⁽¹⁾, 0< z < d,

ψ_j⁽²⁾, −h < z <0, forj = 1,3, and φ₂ =







ψ₂⁽¹⁾, 0< z < d,

ψ₂⁽²⁾, −a < z < 0, (4.1) with the superscripts denoting the specific layer.

By taking into account all information, the boundary value problems for φ_j, j = 1,2,3 are found to satisfy modified Helmholtz equation

(∇²_x,z−k_y²)φ_j = 0. (4.2)

With the help of equations (4.1), (4.2) can be written as follows:

(∇²_x,z−ky2

)ψ_j⁽¹⁾ = 0, 0< z < d, (∇²_x,z−k_y²)ψ_j⁽²⁾ = 0, −h < z < 0,

)

forj = 1,3, (4.3)

87

(∇²_x,z−k_y²)ψ₂⁽¹⁾ = 0, 0< z < d, (∇²_x,z−k_y²)ψ₂⁽²⁾ = 0, −a < z <0.

)

(4.4)

For Regions 1 and 3, the linearized mean free surface condition has the following form:

∂ψ_j⁽¹⁾

∂z −Kψ_j⁽¹⁾ = 0 onz =d (j = 1,3). (4.5) However, the linearized mean free surface condition for Region 2 differs since the region is porous and is given by

∂ψ₂⁽¹⁾

∂z −Kγψ₂⁽¹⁾ = 0 onz =d, (4.6)

whereγ =m+ if denotes the complex porous impedance parameter. In Appendix A, the details are provided for equations in a porous medium, as found in Dalrymple et al. [25].

Further, Region 2is assumed to have a porosity .

It is assumed that there exists no cavitation between the elastic bottom and the lower layer water surface. We obtain the linearized boundary condition on the mean elastic bottom forφ_j, j = 1,3as

EˆI ∂⁴

∂z⁴ −ρp~ω²+ρ2g ∂φ_j

∂z −ρ2ω²φj = 0 onz =−h. (4.7) This can also be written explicitly as

E∂⁵ψ_j⁽²⁾

∂z⁵ + (1−δK)∂ψ⁽²⁾_j

∂z −Kψ⁽²⁾_j = 0, (4.8)

whereE = ˆEI/ρ₂g with EIˆ being the flexural rigidity of the elastic plate, whereEˆ is the Young’s modulus,I =~³/(12(1−ν))andν,~and ρ_p are, respectively, the Poisson’s ratio, the thickness and the density of the elastic plate,g is the usual gravitational constant and δ=ρ_p~/ρ₂.

Due to the rigid foundation, the impermeable bottom condition in Region 2 yields

∂ψ₂⁽²⁾

∂z = 0atz =−a. (4.9)

Let us denote by ρ=ρ₁/ρ₂ the ratio of the densities of the two fluids. Subsequently, the linearized conditions atz = 0 take the following forms:

∂ψ⁽¹⁾_j

∂z = ∂ψ_j⁽²⁾

∂z forj = 1,2,3, (4.10)

ρ ∂ψ⁽¹⁾_j

∂z −Kψ⁽¹⁾_j

!

= ∂ψ⁽²⁾_j

∂z −Kψ⁽²⁾_j

!

forj = 1,3, (4.11)

ρ ∂ψ⁽¹⁾₂

∂z −Kγψ⁽¹⁾₂

!

= ∂ψ⁽²⁾₂

∂z −Kγψ⁽²⁾₂

!

. (4.12)

Thereafter, combining the potentials ψ_j⁽¹⁾ and ψ_j⁽²⁾ intoφ_j as given in (4.1), the matching conditions across the wall yield

φ₁ =γφ₂,

∂φ₁

∂x =∂φ₂

∂x ,







atx= 0, (4.13)

φ3 =γφ2,

∂φ₃

∂x =∂φ₂

∂x ,







atx=B. (4.14)

The above conditions arise due to the continuity of vertical velocity component and the pressure at x= 0 and x=B.

Due to the rigid step, the velocity potentials ψ₁⁽²⁾ andψ⁽²⁾₃ satisfy the following bound- ary condition:

∂ψ_j⁽²⁾

∂x = 0 atx= 0 for −h < z <−a, forj = 1,3. (4.15) Some additional constraints are required to be imposed at the edges of the elastic bottom. Practically, the rigid base of the structure is being constructed upon the bottom and that is why the clamped edge condition is justified to be assumed with respect to the bottom as an elastic plate. Furthermore, the edge conditions are also helpful in view of the need for stable creation of such structures. Under the physical considerations of our model, it is assumed that the elastic plate edges satisfy the clamped-edge boundary conditions at(x, z) = (0,−h),(B,−h)which are as follows:

∂²ψ_j⁽²⁾

∂x∂z = 0, ∂ψ_j⁽²⁾

∂z = 0 forj = 1,3. (4.16)

There exist some more standard edge conditions which can be applied separately at the edge, e.g., the simply supported edge boundary conditions which are given by

∂³ψ_j⁽²⁾

∂x²∂z = 0, ∂ψ_j⁽²⁾

∂z = 0 forj = 1,3. (4.17)

It can be added here that some more complicated boundary conditions may be taken into account. However, in the present context, such a situation does not arise because of

4.1. Scattering by the porous structure 89