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

• Ship surging and surf-riding under multi-frequency wave excitation was investigated.
• Lagrangian Coherent Structures were identified via finite-time Lyapunov exponent fields.
• These structures delineate the skeletal structure of the phase-flow.
• The identification procedure was combined with a feature tracking method.
• Critical points of the acceleration field seem to correlate with surf-riding events.

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.

Ship trajectory in bi-chromatic waves (black line), hyperbolic Lagrangian Coherent Structures at selected time instants (six sections) and paths of three critical points of the acceleration field (blue and red lines)Figure optionsDownload as PowerPoint slide