1.7. Outline of the thesis 27
the various parts of the structure is carried out. It is worth noting that, when an oblique wave strikes on the structure consisting of two different porous blocks, reflection as well as waveload present significant differences as compared to the hydrodynamic coefficients found by scattering by the same porous blocks. The analytical results are compared with an established result and a good agreement is found. Moreover, the porous bottom is considered for a similar kind of problem and a comparison is made.
Chapter 3 investigates the oblique water wave interaction by a double layer compos- ite caisson type breakwater placed upon a rigid base, with a free surface and a porous sea-bed. For dealing with the trapping problem, a rigid sea-wall is considered at a finite distance from the breakwater. A structure with two different porous blocks results in better reflection, transmission as well as waveload than a single block structure. The ana- lytical results are compared with an established result and a good agreement is observed.
Periodic pattern of trapping positions is also observed. Reasonable effect on bottom porosity can be observed in the hydrodynamic coefficients. Moreover, a thin perforated wall in front of the porous structure is considered which establishes the effectiveness of the structure.
Chapter 4 deals with the water wave interaction by a single block caisson structure with a perforated front wall upon a rigid base in a two-layer fluid bounded above by a free surface and below by an elastic bottom. Different velocity potentials are explored as previously and the relevant dispersion relation is fully analysed. The mode swapping in a porous body is discussed. The matching eigenfunction expansion method and linear algebraic approach are used to get the entire analytical solution along with the unknown coefficients. The reflection and transmission coefficients, as well as the waveload, free surface and interface elevations, shear force etc. are computed numerically. The stiffness of the elastic bottom is also examined. Computed results are compared with single layer and two-layer interaction problems. The impact of bottom elasticity is discussed for hydrodynamic coefficients.
Chapter 5 deals with an identical water wave interaction by multiple block caisson structure with a perforated front wall upon a rigid base in a two-layer fluid bounded above by a free surface and below by an elastic bottom. As discussed in Chapter 4, the matching eigenfunction expansion method is used to get the entire analytical solution along with the unknown coefficients. The reflection coefficients, as well as the waveload, free surface and interface elevations, shear force etc., are computed numerically. The stiffness of the elastic bottom is examined. The inclusion of a perforated front wall has the following effects: (i) the percolative property of the front wall increases the effectiveness of the breakwater in generating lower waveload effective on the rigid wall, (ii) significant difference in the reflection coefficients in both cases.
Chapter 6 discusses the water wave interaction in a two-layer fluid with a thin poro- elastic structure placed upon a porous bottom. A partially reflective sea-wall is considered
1.7. Outline of the thesis 29 at a distance. The dispersion relation is analysed deeply, and mode-swapping as well as other characteristics are discussed. The dead water analogue is also illustrated. The en- ergy identity for such a complex system is also derived. Various hydrodynamic coefficients are compared with different values of parameters like the porous-effect parameter and the length of the barrier, the angle of incidence, and the porosity of the sea-bed etc. Trap- ping is analysed for the occurrence of wave reflection minima. Due to adequate change of structural parameters of the barrier, a significant portion of the wave energy dissipation can be observed.
An identical experiment is carried out in Chapter 7, but this time the poro-elastic structure is substituted with a thin porous body. Along with the other issues inChapter 6, here the water wave radiation is also explored. When the propagating wave strikes the submerged vertical porous barriers, significant variations in the porous-effect parameter of the sea-bed show a significant impact. A reasonable change in the porous-effect parameters in the structure has an important impact in wave radiation. Furthermore, owing to energy absorption by such a geometry, the important energy identities are also established.
Chapter 8 studies the oblique surface wave interaction with a porous structure in front of a moored floating elastic plate in a fluid consisting of two immiscible layers. The following issues are examined: (i) incident and reflected wave scattering, (ii) structural waveload, (iii) plate elevation, bending moment and shear force, (iv) effect of mooring lines on the plate. Comparison of moored edge emphasizes that suitable edge conditions must be chosen in order to achieve optimum panel deflection and other physical quantities.
An optimal structural length is proposed so that maximal wave reflection and minimal transmission can be obtained. Furthermore, the lowest waveload on the plate can be obtained from appropriate values of porous structures.
Finally, Chapter 9 contains a concise review of the findings in this thesis, focusing on the most relevant outcomes. It also gives details on the scope of future investigations.
CHAPTER 2
Linear water wave interaction with a composite porous structure in a two-layer fluid flowing over a step-like sea-bed
2.1 Mathematical formulation
The problem of surface wave interaction with a two-block porous structure, possessing block-wise distinct porosity, placed on an impermeable multi-step bottom is considered in Cartesian coordinates. Using linear water wave theory, irrotational and simple harmonic motion are assumed in an inviscid, incompressible fluid of two layers with a free surface and a linear interface under the action of gravity. Any effect due to surface tension at the interface of the two fluids is ignored since, for such cases, the contribution of surface tension is found to be negligible. The upper layer fluid having density ρ1 has its free surface at z = d and the lower layer fluid with density ρ2 has its lower boundary as an
Figure 2.1: Definition sketch of wave interaction with porous blocks placed on an elevated bottom
impermeable step-bottom. The surface wave is obliquely incident on the porous structure, which has its upper boundary coinciding with the mean free surface, at an angle θ to the x-axis. This ensures that velocity potentials Φj, j = 1,2,3 can be taken in the form Φj(x, y, z, t) = Re{φj(x, z)ei(kyy−ωt)}, where ω is angular wave frequency, ky = k1,Isinθ is the y-component of the wavenumber k1,I of the plane progressive wave propagating in the surface mode where the subscripts j and n in kj,n correspond to the j-th region and n-th wavenumber, respectively, Re denotes the real part ofφj(x, z)ei(kyy−ωt) and i =√
−1 is the usual imaginary quantity. The fluid region is split into three regions as follows:
Region 1 (−∞ < x < 0, −h1 < z < d), Region 2 (0 < x < L, −h2 < z < d) and Region 3 (L < x < L+D, −h3 < z < d). The contribution of ky leads to Snell’s law for refraction across discontinuities in the water depth which obeys the equality of ky across all three regions. Therefore, the variation of the potential functions φj( j = 1,2,3) in the y-direction is the same so that we are able to match the velocity potentials along the vertical boundaries. Here, φj denotes the the velocity potential in the j-th sub-domain, j = 1,2,3, in Regions 1, 2 and 3, respectively, as shown in Figure 2.1. With all the above considerations, the boundary value problem for each potential φj is governed by the modified Helmholtz equation
(∇2x,z−k2y)φj = 0 for j = 1,2,3, (2.1) in which ∇2x,z = ∂2
∂x2 + ∂2
∂z2 is the two-dimensional Laplacian operator.
The linearized boundary condition at the mean free surface for Region 1 is
∂φ1
∂z −Kφ1 = 0 on z =d, (2.2)
where K = ω2 g .
On the other hand, the linearized boundary condition at the mean free surface in Regions 2 and 3 are
∂φj
∂z −Kγj−1φj = 0 on z =d (j = 2,3). (2.3) Derivation of mean surface boundary condition (2.3) was detailed for homogeneous poros- ity in a homogeneous fluid by Das and Bora [33]. The impermeable step bottom has the boundary condition
∂φj
∂z = 0 on z =−hj, for j = 1,2,3. (2.4)
Considering the ratio of the densities of the two fluids asρsuch that0< ρ=ρ1/ρ2 <1,