Coherent structures in phase space, governing the nonlinear surge motions of ships in steep waves

Abstract

In steep multi-chromatic seas, ship surge dynamics can become intricate and the full variety of exhibited motions is unknown. This accrues, partly, from the nonlinear nature of surge motion; and partly because, for multi-frequency waves, the phase-space flow of the dynamical system becomes time-dependent. Accordingly, conventional concepts that were applied in the past for analyzing stationary phase-space flows are rendered incapable to support in-depth exploration of ship dynamics. Towards overcoming this limitation, use of the concept of hyperbolic Lagrangian Coherent Structures (LCSs) is proposed. These phase-space objects can be regarded as the “finite-time” generalizations of the stable and unstable manifolds of hyperbolic fixed points defined in “time-invariant” dynamical systems. They can be described as, locally, the strongest repelling or attracting material surfaces (curves in the case of 2-dimensional systems) advected with the phase flow. We have identified hyperbolic LCSs that are innate to the phase-flow associated with the surge motion of a ship in astern seas. To the global approach of LCS identification, a supplementary computational scheme is incorporated, aiming to track, in space-time, local “features” of the flow, connected with surf-riding. The emerging toolset can enhance current efforts towards a rigorous assessment of ship dynamic stability in following seas.