Interaction of a Second-Order Solitary Wave With an Array of Four Vertical Cylinders

Abstract

In this paper the interaction of a plane second order solitary wave with an array of four vertical cylinders is investigated. The fluid is assumed to be incompressible and inviscid. The diffraction analysis assumes irrotationality, which allows for the use of Boussinesq equations. A simultaneous expansion in a small nonlinearity parameter (wave amplitude/depth) and small dispersion parameter (depth/horizontal scale) is performed. Boussinesq models, which describe weakly nonlinear and weakly dispersive long waves, are characterized by the assumption that the nonlinearity and dispersion are both small and of the same order. An incident plane second order solitary wave is the Laitone solution of Boussinesq equations. The representation of variables as the series of small nonlinearity parameters leads to the sequence of linear boundary value problems of increasing order. The first order approximation can be determined as a solution of homogeneous differential equations and the second order approximation follows as a solution of non-homogeneous differential equations, where the right hand sides may be computed from the first order solution. For the case of a single cylinder an analytical solution exists. However, when dealing with more complex cylinder configurations, one has to employ numerical techniques. In this contribution a finite element approach combined with an appropriate time stepping scheme is used to model the wave propagation around an array of four surface piercing vertical cylinders. The velocity potential, the free surface elevation and the subsequent evolution of the scattered field are computed. Furthermore, the total second order wave forces on each individual cylinder are determined. The effect of the incident wave angle is discussed.