Analytical Predictions of the Air Gap Response of Floating Structures

Two separate studies are presented here that deal with analytical predictions of the air gap for floating structures. (1) To obtain an understanding of the importance of firstand second-order incident and diffracted wave effects as well as to determine the influence of the structure’s motions on the instantaneous air gap, statistics of the air gap response are studied under various modeling assumptions. For these detailed studies, a single field point is studied here – one at the geometric center (in plan) of the Troll semi-submersible. (2) A comparison of the air gap at different locations is studied by examining response statistics at different field points for the semi-submersible. These include locations close to columns of the four-columned semi-submersible. Analytical predictions, including firstand second-order diffracted wave effects, are compared with wave tank measurements at several locations. In particular, the gross root-mean-square response and the 3-hour extreme response are compared. BACKGROUND The air gap response, and potential deck impact, of ocean structures under random waves is generally of considerable interest. While air gap modeling is of interest both for fixed and floating structures, it is particularly complicated in the case of floaters because of their large volume, and the resulting effects of wave diffraction and radiation. These give rise to two distinct effects: (1) global forces and resulting motions are significantly affected by diffraction effects; and (2) local wave elevation modeling can also be considerably influenced by diffraction, particularly at locations above a pontoon and/or near a major column. Both effects are important in air gap prediction: we need to know how high the wave rise (step 2), and how low the deck translates vertically (due to net heave and pitch) at a given point to meet the waves. Moreover, effects (1) and (2) are correlated in time, as they result from the same underlying incident wave excitation process. We focus here on analytical diffraction models of air gap response, and its resulting stochastic nature and numerical predictions under random wave excitation. Attention is focused on a semi-submersible platform, for which both slow-drift motions (heave/pitch) and diffraction effects are potentially significant. This air gap response presents several new and interesting challenges. It is the first response limit state where we need to simultaneously include both second-order sumand difference-frequency effects (on the wave surface), and second-order differencefrequency effects (on slow drift motions and generally, on the wave surface as well). The sumand differencefrequency waves and the difference-frequency heave and pitch motions can both influence the air gap. The air gap response is further complicated because the heave, pitch, and roll motions of the floating structure are generally coupled. Moreover, the motions and the net wave elevation, both of which affect the air gap, are correlated in time as they result from the same underlying incident wave excitation process. Note that air gap modeling has been the subject of previous work within the Reliability of Marine Structures Program at Stanford University. For example, Winterstein and Sweetman (1999) apply a fractile-based approach to develop a scaling factor between the statistics of the incident waves and those of the associated air gap demand. Results are shown here from frequency-domain analyses which permit careful study and isolation of various effects: e.g., wave forces on a fixed (locked-down) structure, the effect of structural motions on air gap response, and finally, the effect of different local wave elevation models. For reference, a complete second-order diffraction model is formulated and studied. Compared with this complete model, various simplified models are imposed and evaluated: (a) second-order wave elevation effects are neglected completely, or (b) these second-order effects are approximated by analytical Stokes theory, which retains second-order effects on the incident wave but neglects second-order diffraction. Use of (a) and (b) would significantly simplify the analysis, avoiding the costly step of second-order diffraction. The local wave modeling in step (b) is found to be quite important in predicting the air gap response of two semi-submersibles studied here. THEORY Volterra Series and Transfer Functions In modeling nonlinear systems, such as floating structures, it is common to employ Volterra series that permit one to describe the response (output) of such systems. The nonlinear system is defined in terms of firstand second-order transfer functions. For floating structures, these transfer functions are obtained from firstand second-order wave diffraction analysis programs (e.g., WAMIT, 1995). In order to study the response of a floating structure to random seas, we start by defining an irregular sea surface elevation, η(t), as a sum of sinusoidal components at N distinct frequencies: ) exp( ; ) ( 2 where ; ) exp( Re ) cos( ) (