Hydrodynamic Model Induced Differences in SPM Post Pitchfork Bifurcation Paths

Abstract

Several models of the hydrodynamic forces acting on a ship hull in maneuvering have been developed in the last 50 years. These models make possible analysis of ship maneuverability in high and low speeds. Following Bernitsas et al. [1], such hydrodynamic models may be classified into two major schools: the hydrodynamic derivatives (HD) models (first school) and “cross-flow” models (second school). The former is based on Taylor series expansion of the forces while the corresponding coefficients are determined experimentally and remain velocity independent for relatively low velocities. The second school heuristically combines short-wing theory (Jones) and cross-flow experimental data. The aim of this work is to establish and review a certain discrepancy observed in post pitchfork bifurcation paths depending on which school of modeling is adopted. This discrepancy exists in the practical problem of a Single-Point Mooring (SPM) system in a steady ocean current. This discrepancy appears immediately after the point of pitchfork bifurcation of the equilibrium yaw angle versus the longitudinal position of the line attachment point on the hull. According to HD models (e.g., Abkowitz [2]) such a bifurcation curve is a square-root post pitchfork path (e.g., Papoulias and Bernitsas [3]) while cross-flow models (e.g. Leite et al. [4]) predict a different shape of this path at the onset of the post bifurcation curve. Although the practical effect of such a discrepancy may be negligible for SPM systems, this is valuable in assessing an important difference in the distinct approaches followed by the hydrodynamic schools of modeling. Specifically, viscous forces are modeled by odd nonlinear terms in velocity, which are bilinear in the cross-flow models and cubic in the HD models. In this work, experimental results on the aforementioned post pitchfork bifurcation paths are presented and the origin and relevance of the observed discrepancy are discussed. Finally, results presented by Hooft [5] show that yaw angle dependence on bilinear velocity terms regarding cross-flow coefficients would be necessary for a more precise representation of bifurcation patterns near the pitchfork bifurcation. Such patterns may be strongly influenced by hull form.