A New Numerical Tool for Fast Ships in Following Seas

The six-degrees-of-freedom ship motions of a ship at speeds other than zero are always measured in terms of encounter frequency, and often, the incident waves in experimental data are also measured only in the encounter frequency domain. Using these measured data to obtain transfer functions from irregular following sea ship motions is complicated by the combined effects of very low encounter frequencies and the “folding” of the sea spectra. This results in having both overtaking and encountered waves of the same encounter frequency but different wavelengths. Computing transfer functions becomes untenable when the ship speed approaches the wave phase velocity, where the encounter spectrum has a mathematical singularity. St. Denis and Pierson (1953, “On the Motions of Ships in Confused Seas,” Soc. Nav. Archit. Mar. Eng., Trans., 61, pp. 280–357) suggested the basic relationships between response ship motions or moments that can be developed in the wave frequency domain at the outset. The St. Denis–Pierson method is based on a linear theory and works well when the ship response regime is linear or weakly nonlinear. However, for high-speed craft operating at different headings where the problems are nonlinear, especially strongly nonlinear, the St. Denis–Pierson assumptions will break down inducing error (1953, “On the Motions of Ships in Confused Seas,” Soc. Nav. Archit. Mar. Eng., Trans., 61, pp. 280–357). Furthermore, using the frequency resolution method to remove the singularity point may also induce errors, especially when the singularity point is located near the peak of stationary frequency. How to obtain the correct frequency resolution in the local region of singularity point is still an unsolved problem. In this study, we will propose a new method capable of predicting ship response motions for crafts with nonlinear or strongly nonlinear behaviors quantitatively. For example, using this method, one can use measured ship motion data in head seas to predict the motions of the ship at high speed in following seas. The new method has six steps, including using a filter to eliminate those unexpected modes that are not from incident waves, inertial motions, or nonlinear interactions, and applying a higher-order Taylor expansion to eliminate the singularity point. We refer to the new method as the Lin–Hoyt method, which agrees reasonably well with computations of the nonlinear “digital, self-consistent, ship experimental laboratory ship motion model,” also known as DiSSEL (2006, “Numerical Modeling of Nonlinear Interactions Between Ships and Surface Gravity Waves II: Ship Boundary Condition,” J. Ship Res., 50(2), pp. 181–186). We also use experimental head sea data to validate the simulations of DiSSEL. The Lin–Hoyt method is fast and inexpensive. The differences in the results of the numerical simulations obtained by the Lin–Hoyt method and other linear methods diverge rapidly with increased forward ship speed due to the nonlinearity of ship motion responses.