Elsevier

Applied Ocean Research

Volume 29, Issue 4, November 2007, Pages 180-190
Applied Ocean Research

Interaction of regular waves with a group of dual porous circular cylinders

https://doi.org/10.1016/j.apor.2008.01.004Get rights and content

Abstract

Diffraction of linear waves around a group of dual porous cylinders consisting of a thin and porous outer cylinder with an impermeable inner cylinder is investigated analytically based on the eigenfunction expansion method proposed by Spring and Monkmeyer [Spring BH, Monkmeyer PL. Interaction of plane waves with vertical cylinders. In: Proceedings 14th international coastal engineering conference. 1974. p. 1828–47] and further modified by Linton and Evans [Linton CM, Evans DV. The interaction of waves with arrays of vertical circular cylinders. Journal of Fluid Mechanics 1990;215:549–69]. The present formulation is an extension of the work of Wang and Ren [Wang KH, Ren X. Wave interaction with a concentric porous cylinder system. Ocean Engineering 1994;21(4):343–60], wherein; the interaction of linear waves with a single concentric porous cylinder system was studied. This paper aims at investigating the influence of multiple interactions between the cylinders in the group on the hydrodynamic wave forces, wave run-up and free-surface elevation in their vicinity. Further, the study focuses on the variation of the forces and run-up on the individual cylinders within the group compared to that on isolated cylinders.

Introduction

Studies on the interaction between ocean waves and groups of cylinders have become an increasingly important topic due to the advances in the design and construction of large offshore structures for the exploitation of large reserves beneath the sea bed. Considerable research has been made in the estimation of hydrodynamic loads on an isolated cylinder as well as on the interaction of cylinders in a group. When a wave is incident on a group of cylinders, it is necessary to consider not only the diffraction due to each of the cylinders, but also the multiple scattering due to the presence of neighboring cylinders within the group. The above-stated phenomena lead to complexities in the computation of the hydrodynamic loading and wave run-up on the individual cylinders.

Spring and Monkmeyer [1] have proposed an analytical solution for the diffraction of linear waves by an array of bottom-mounted, surface piercing impermeable circular cylinders using the eigenfunction expansion approach. Simon [4] developed a plane-wave approach for a uniformly spaced linear array of axi-symmetric bodies. Considering a large-spacing approximation, whereby replacing the diverging waves on one body due to the scattering of another body by a single plane wave, a simplified system was obtained. McIver and Evans [5] extended the above approach by including a correction term in the plane-wave approximation. Kagemoto and Yue [6] have proposed an analytical solution for the general three-dimensional water–wave diffraction problem. This method also includes the interaction of evanescent waves, which is solved exactly in the context of linearized theory. Linton and Evans [2] simplified the theory of Spring and Monkmeyer [1] leading to the direct evaluation of the wave forces and run-up on the cylinders. Other salient works on the interaction of waves with impermeable circular cylinders are included in a detailed review made by McIver [7].

Most of the earlier studies assume that the cylinders are impermeable. Limited amount of work has been done in the context of wave diffraction by thin-walled porous cylindrical structures. Wang and Ren [3] were the earliest to study the wave interaction with a concentric surface piercing porous outer cylinder protecting an impermeable inner cylinder. The free-surface elevation, net hydrodynamic forces and wave-induced over turning moments on both cylinders were determined analytically. It was reported that the existence of outer porous cylinder reduced the hydrodynamic force on the inner cylinder, as compared to, when it is exposed to direct wave impact. The above work was extended for applications like semi-porous cylindrical breakwater by Darwiche et al. [8] and for semi-porous cylindrical breakwater mounted on a storage tank by Williams and Li [9]. Govare et al. [10] have analytically investigated the wave kinematics around a protected impermeable pile. For a set-up consisting of an impermeable cylinder encircled by a porous cylinder Vijayalakshmi [11] measured the wave forces and run-up on the two structures independently through laboratory studies, the results of which were compared with that predicted by the boundary integral method. Silva et al. [12] have developed a numerical scheme using the central difference method to solve a modified-time-dependent mild-slope equation for understanding the phenomena of linear waves propagating over a rapidly changing finite porous bed. The application of the above numerical model to the case of concentric dual circular cylinders proved to be efficient with that of the results of Govare et al. [10]. Based on the method proposed by Linton and Evans [2], Williams and Li [13] investigated theoretically the hydrodynamic forces and run-up on an array of bottom-mounted surface piercing circular cylinders each of which has a porous sidewall. Recently, Umnova et al. [14] have studied the effects of porous covering on sound attenuation by periodic arrays of cylinders.

In this paper, the complete analytical model with the inclusion of multiple interactions between the dual cylinders in a group is presented. The outer cylinder is considered thin and porous. The force, run-up on the inner and outer cylinders, and the wave elevation in the vicinity of the cylinders under the interaction with regular waves with normal angle of incidence are presented and the analyses of the results are discussed herein. The effects of the porosity of the outer cylinder and the spacing between the inner and outer cylinders on the forces and run-up on the cylinders have been brought out. The results from the proposed model have been compared with the available experimental results.

Section snippets

Formulation of the problem

The definition sketch of an arbitrary group of N bottom-mounted surface piercing, dual porous cylinders in a constant water depth, h is shown in Fig. 1. The global coordinates (x,y,z) are defined with the origin located at the still water level and z pointing vertically upwards. The radius of the jth outer cylinder is bj and that of the corresponding inner cylinder is aj. The center of each cylinder is taken as the origin of the local polar coordinate system (rj,θj), j=1,2,,N. The various

Analytical solution

The total potential in the outer region ϕ1 is given by ϕ1(x,y)=ϕinc(x,y)+j=1Nϕs(j) where, ϕinc(x,y)=eikrcos(θβ) is the incident potential and, in the jth polar coordinate system, the incident potential can be expressed by, ϕinc(j)=Ijn=Jn(krj)ein(π/2β+θj) where Ij=ei(xjcosβ+yjsinβ) is the phase vector associated with jth cylinder and Jn ( ) denotes Bessel function of the first kind of order n, β is the wave direction.

In the case of multiple cylinders, the wave scattered by jth cylinder may

Wave forces and surface elevations

Based on the derived velocity potentials, the wave excitation forces and run-up on the outer and inner cylinders as well as the water surface elevation around the group of cylinders can be computed. The exciting forces in the x-direction on the outer and inner cylinders are obtained by the integration of the pressure on the surface of the respective cylinder, FOx(j)=iρgHk2tanh(kh)S1Z1jS1J1(kbj)+i2G0πkbjJ1(kaj)(A1jA1j)FIx(j)=ρgHk2tanh(kh)2G0πkbjZ1jS1J1(kbj)+i2G0πkbjJ1(kaj)(A1jA1j).

The

Results and discussion

A computer program using MATLAB has been developed to implement the above analysis, and the diffraction characteristics for several configurations have been studied. It is found from the literature (Linton and McIver [18]) that by adopting the order of the Bessel functions (M) as 6, sufficient accuracy can be obtained for engineering applications. The efficiency of the proposed formulation has been verified with that of the past works for various limiting cases. The details of the numerical

Summary

In the present study, the wave interaction with a group of dual porous cylinders consisting of an outer porous cylinder protecting an impermeable inner cylinder is investigated using the eigenfunction expansion approach. The hydrodynamic forces on the individual cylinders are derived considering the multiple interactions between the cylinders. The results of the limiting cases were compared with the past experimental and analytical studies. A detailed parametric study has been carried out based

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