A four-quadrant model for ship maneuvering in forward and backward motions

This study developed a four-quadrant maneuvering model that extended the Abkowitz-type model by explicitly incorporating the propeller rotation rate as a control input. Ship motions were classified into four operational quadrants: forward (Q1), crash forward (Q2), backward (Q3), and crash stop (Q4). Propeller thrust was modeled using measured open-water characteristics, while lateral forces induced by the unidirectional propeller rotation were derived from experimental data. A single-screw, twin-rudder inland waterway bulk carrier operating on the Rhine River served as the case study vessel. Systematic numerical captive model tests, employing Reynolds-averaged Navier–Stokes (RANS) equations-based simulations, were conducted to determine the hydrodynamic coefficients across all quadrants. The proposed model was validated against full-scale trials encompassing diverse maneuvers such as acceleration, stopping, turning, and zigzag tests. Additionally, the model’s predictive capability was evaluated for stopping, turning, and zigzag maneuvers in both forward and backward motions. The results demonstrated the model’s accuracy and robustness in capturing ship dynamics across a wide range of operational scenarios.