On the complementarity of the slender-body and Newman’s approximations for difference-frequency second-order wave loads on slender cylinders

Abstract

As reported in the literature, Newman’s and the slender-body approximations for second-order wave loads (the latter often referred to as Rainey’s equation) present an interesting complementarity for the case of a bottom mounted vertical cylinder, indicating that each one captures different physical aspects of the problem. In light of that result, the present work aims at analyzing this complementarity by inspecting the different components that constitute the second-order wave loads, including the ones due to body motions, and by relating them to the incident and diffracted waves. This is done by analyzing the case of a single surface piercing cylinder under the action of long-crested bichromatic waves, first considered to be bottom mounted and then in floating condition. It is also shown that the second-order terms from Rainey’s formulation are indeed relevant, reminding one that the common practice of analyzing second-order loads by simply applying Morison’s equation with second-order wave kinematics is not strictly correct.In order to include the effects of body motions, Rainey’s equation, which can be seen as an extension of the inertial part of Morison’s equation to include nonlinear terms, is combined with Pinkster’s approach for the low-frequency second-order loads on floating bodies. This combination is made in order to allow the evaluation of the forces considering the mean body position, in opposition to the original approach that considers the instantaneous body position, allowing an IFFT algorithm to efficiently compute second-order wave loads in a real sea condition directly in time domain. As a drawback of the slender-body approximation, end effects due to the extremities of the cylinder and effects due to wave scattering and radiation are lost, as is the case when using Morison’s equation to evaluate first-order loads, and the question is to what extent they can be neglected when dealing with second-order wave loads.