The effect of short-crested wave phase on a concentric porous cylinder system in the wind blowing open sea

Cylindrical structures are widely employed in the offshore engineering. How to protect the structures in real sea environment is a vital issue for scientists and engineers. In this paper, the interaction of short-crested waves with a concentric surface-piercing two-cylinder system is studied based on linear wave theory using the scaled boundary finite-element method (SBFEM), a novel semi-analytical method with the advantages of combining the finite-element method (FEM) with the boundary-element method (BEM). The interior cylinder is impermeable and the exterior cylinder is thin and porous to protect the interior cylinder. Both of cylinders are bottom mounted. Wave elevation in the annular region is found dependent on the short-crested phase, and the total forces fluctuate as the phase changes. It is further revealed that the maximum total forces are smaller than those induced by plane waves and standing waves with the same total wave number. Introduction In order to reduce the direct wave impact, coastal and offshore structures are often constructed with one or more protective porous layers. Examples are rock-filled porous breakwaters outside harbours, concentric porous outer protective structure with the main structure in its interior. One such application is the successful Ekofisk gravity offshore structure in the North Sea. For these reasons, wave motion through a porous structure has attracted considerable interests among researchers in coastal and ocean engineering. In addition to the porous wavemaker theory of Chwang [1] and subsequent works, investigations on waves past a porous structure are primarily been concentrated on the hydrodynamic effects of a porous structure on the incoming wave trains, or wave impact on porous structures as a breakwater in a harbour (e.g., [13, 14]). In most cases, Darcy’s law for a homogeneous porous medium has been applied. Yu and Chwang [13] investigated the resonance in a harbour with porous breakwaters with the wave entering at an arbitrary angle. Yu and Chwang [14] performed extensive study on the transmission characteristics of waves past a porous structure. The wave behaviour within the porous medium was also investigated. It was found that there is an optimum thickness for a porous structure beyond which any further increase of the thickness may not lead to an appreciable improvement in reducing its transmission and reflection characteristics. Wang and Ren [9] also studied the performance of a flexible and porous breakwater. Additional related work can be found in the review article of Chwang and Chan [2]. Though considerable research efforts have been devoted to the wave interaction with porous structures, relatively limited attention has been focused on the wave diffraction by a concentric bottom-mounted porous cylindrical structure, where the interior cylinder is impermeable and the exterior cylinder is thin and porous. Wang and Ren [10] investigated analytically the plane wave diffraction by the above-mentioned system. They found that hydrodynamic forces on the interior cylinder as well as wave amplitudes around the windward side of the interior cylinder are reduced compared to the case of a direct wave impact on the interior cylinder. As the annular spacing increases, the hydrodynamic force on the interior cylinder decreases. It was further shown that, as the porosity of the exterior cylinder increases, the hydrodynamic force on the interior cylinder increases. Darwiche et al. [3] also studied the wave diffraction by a two-cylinder system, with the exterior cylinder being porous only in the vicinity of free surface. Williams and Li [11] further extended the work by mounting the interior cylinder on a storage tank. The aforementioned studies on ocean surface waves interaction with a vertical porous cylindrical structure are generally two-dimensional. In reality, however, the ocean waves are more complex, and better described by three-dimensional shortcrested waves. They also commonly arise, for example, from the oblique interaction of two travelling plane waves or intersecting swell waves, from the reflection of waves at non-normal incident off a vertical seawall, as well as from diffraction about the surface boundaries of a structure of finite length. Such waves are of paramount importance in coastal and offshore engineering design. Unlike the plane waves propagating in a single direction, and the standing waves fluctuating vertically in a confined region, short-crested waves can be doubly periodic in two horizontal directions, one in the direction of propagation and the other normal to it [8]. Theoretical analysis on short-crested waves interaction with a vertical cylinder can be found in [15, 16]. Zhu [15] presented an analytic solution to the diffraction problem for a circular cylinder in short-crested waves using linear potential wave theory and revealed that the pressure distribution and wave run-up on the cylinder were quite different from those of plane incident waves. Their patterns become very complex as ka (i.e., total incident wave number k times cylinder radius a) becomes large. The hydrodynamic forces on the cylinder become smaller as the short-crestedness of the incident wave increases. Subsequently, Zhu and Moule [16] observed that the hydrodynamic force induced by short-crested waves varies with the phase angle perpendicular to the direction of wave propagation. Recently, a semi-analytical method, called scaled boundary finite-element method (SBFEM) for solving linear partial differential equations has found successful application to soilstructure interaction problems. The SBFEM method was proposed by Song and Wolf [5] and systematically described by Wolf [12]. Combining the distinct advantages of the finiteelement and boundary-element methods, only the structure boundary is discretised with surface finite-elements. This, in turn, transforms the governing partial differential equations to a set of ordinary differential equations, and solves them analytically. The method represents singularities and unbounded domains accurately and efficiently when compared to the complete finite-element method and requires no fundamental solution as needed by the boundary-element method. Fewer elements are required to obtain very accurate results [12]. Li et al. [4] solved the problem of plane wave diffraction by a vertical cylinder using SBFEM. Similar to the approach of Wolf [12] in obtaining a solution for soil-structure interaction, Li et al. [4] adopted an algebraic series to obtain the final solution. However, for low frequency waves with ka¿ 1 the series converges very slowly, hardly approaching to the exact solution. Tao et al. [6] applied the SBFEM to solve short-crested waves interaction with a circular cylinder. Instead of using a power series, Tao et al. [6] chose Hankel function to solve the Helmholtz equation in the unbounded domain. The radial differential equation is solved fully analytically in all frequency ranges. Without relying on any other numerical schemes, the semi-analytical model for the plane-wave diffraction by a single circular cylinder is shown to reproduce the analytical solution for all the physical properties including wave run-up, effective inertia and drag coefficients, and total force very accurately and at very low computational cost. Although much effort has been made on wave interaction with porous cylinders and breakwaters, works on the short-crested waves interaction with a concentric porous cylindrical structure, especially the effects of short-crested phase have been relatively few. This paper aims at this particular aspect in a quantitative manner. It will apply the SBFEM model developed in [6] to study the short-crested wave interaction with a concentric porous cylindrical structure. Mathematical Formulation Consider a monochromatic short-crested wave train propagating in the direction of the positive x axis. A structure consisting of two concentric fixed vertical cylinders extend from the sea bottom to above the free surface of the ocean along z axis. The origin is placed at the centre of the cylinders on the mean water surface (see Fig. 1). The exterior cylinder is made porous and the interior cylinder is impermeable. The whole fluid region is divided into two regions, the annular region Ω1 and the region of the outside of the exterior cylinder Ω2. The following notation have been used in the paper: Φ j = total velocity potential, ΦI = velocity potential of incident wave, ΦS = velocity potential of scattered wave, k = total wave number, kx = wave number in x direction, ky = wave number in y direction, ω = wave frequency, h = water depth, A = amplitude of incident wave, a = interior cylinder radius, b = exterior cylinder radius, t = time, ρ=mass density of water, and g= gravitational acceleration. The subscripts j( j = 1,2) denote the physical parameters in the region Ω j( j = 1,2). Assuming the fluid to be inviscid, incompressible and the flow to be irrotational, the fluid motion can be described by a velocity potential Φ j satisfying the Laplace equation ∇2Φ j = 0 in Ω j, (1) subject to the combined free surface boundary condition Φ j,tt +gΦ j,z = 0 at z= 0, (2) and the bottom condition Φ j,z = 0 at z=−h, (3) where the comma in the subscript designates partial derivative with respect to the variable following the comma. The velocity potentials can be decomposed by separating the vertical variable z and the time t from each component as η