Second-Order Hydrodynamic Interactions in an Array of Vertical Cylinders in Bichromatic Waves

Abstract

A complete second-order solution is presented for the hydrodynamic forces on an array of bottom-mounted, surface-piercing, vertical cylinders of arbitrary cross-section in bichromatic waves. Exploiting the constant structural cross-sections, the vertical dependency of the linearized potentials is expressed in terms of eigenfunction expansions. A two-dimensional Green's function approach may then be utilized to solve the first-order problem. Green's second identity is applied to express the second-order forces due to the second-order potential in terms of free-surface and structural integrals involving firs-order quantities and associated linearized radiation potentials obtained by oscillating each of the structures in turn at the sum and difference frequencies of the component first-order waves. Considerable attention is focussed on developing an efficient numerical technique to compute the oscillatory free-surface integral appearing in the second-order force formulation. In the near field the integration is carried out numerically, while in the far-field the integration is carried out analytically utilizing the asymptotic behavior of the potential components. Numerical results are presented for an array of four circular cylinders which illustrate the importance of interaction effects at both the sum and difference frequencies on the second-order hydrodynamic loads on the cylinders.