A NUMERICAL ANALYSIS OF THE WAVE FORCES ON VERTICAL CYLINDERS BY BOUNDARY ELEMENT METHOD

Abstract

With the advances in the design and construction, various types of offshore structures are now commonly constructed with a composite configuration of several legs such as cylinders. Thus, the interaction between waves and multiple cylinders is becoming more and more important. The wave pressure and wave forces acting on the cylinders must be exactly estimate to evaluate the wave effects and the stability of structures. The interaction between waves and offshore structures causes problems such as diffracted waves, reflected waves, and wave forces. Consider N large vertical circular cylinders placed on the bottom. As the incident waves impinge on each cylinder, the reflected waves move outward. On the sheltered side of the cylinders there will be a "shadow" zone where the wave fronts are bent around the cylinders, the so-called diffracted waves. The reflected waves and diffracted waves, combined, are usually called the scattered waves. The scattered waves of each cylinder can affect other cylinders in the group. This process is generally termed diffraction. By the process of diffraction the pressure around the cylinders will be changed and therefore the forces on the cylinders will be influenced. The problems could be dealt with the boundary value with the velocity potential. There have been several studies dealing with the interaction of water waves and multiple cylinders. Twersky (1952) constructed a solution using an iterative procedure in which successive scattered waves by each of the cylinders were introduced at each other. This method was extended to the water wave case by Ohkusu (1974). The main drawback of the iterative procedure is that it rapidly becomes unmanageable as the number of the cylinders increases. Another approach is the direct matrix method, Spring and Monkmeyer (1974) proposed a solution for the interaction of water waves and the cylinders using eigenfunction expansion approach. They formulated the problem in terms of matrix equation and the solution is obtained by the inversion of the matrix. Chakrabarti (1978) extended the work of Spring and Monkmeyer (1974), and obtained the solution for the diffracted wave of multiple cylinders by carrying out the analysis in a complex domain. Subsequently, Linton and Evan (1990) made a major simplification to the theory proposed by Spring and Monkmeyer (1974). On the other hand, there are many studies to solve the diffraction problems of water waves by using boundary element method. Notably, Kim and Park (2007) studied a numerical analysis method to calculate the wave force on isolated vertical circular cylinder by boundary element method and the numerical results of their study are strong agreement with those of MacCamy and Fuchs (1954). Also, Kim and Cao (2008) studied wave forces acting on the two vertical cylinders and three vertical cylinders in water waves. The boundary element method is divided into direct boundary element method and indirect boundary element method. The direct boundary element method derives from the integral equation by Green's or Cauchy's theory. This deriving process is solved by the special application of weighted residual method and it has general integral formulation. Derived integral equation is divided into two parts: the unknown functions and the derivatives of normal direction. One side of them generally has unknown values. The indirect boundary element method is derived integral equation in boundary conditions and is made operation function by singular value of governing equation. The coefficient that is involved operation function is unknown value in direct boundary element. In this paper, a numerical analysis by boundary element method for calculating wave forces acting on multiple cylinders is presented. The numerical analysis method by Green function in direct boundary element method using velocity potential φ is developed. Attention has been concentrated on wave forces on N large vertical cylinders, having radius a and placed on the bottom. To verify this numerical method and to investigate the effect of the neighboring cylinders on the wave forces acting on a cylinder, two-cylinder configuration and three-cylinder configuration are used in this study. The wave forces acting on two vertical circular cylinders and three vertical cylinders obtained from this numerical method are compared with those of Ohkusu (1974) and Chakrabarti (1978). The comparisons show that the computed results of this study are strong agreement with their results. Also in this study, several numerical examples are given to illustrate the effects of various parameters on the wave forces acting on the cylinders such as the cylinder spacing, the wave number, and the incident wave angle. The run-up and free surface elevation around two vertical cylinders and three vertical cylinders are also calculated.