When ships sail in following waves, they often encounter complex nonlinear phenomena. Specifically, variations in waterplane area may lead to a significant reduction in the roll restoring force, and excessive surge forces on the ship can induce unintended acceleration. These phenomena are highly likely to trigger stability failures involving large-angle heel, among which pure loss of stability is a critical concern. In the case of pure loss of stability, when the wave crest remains at midships for an extended period, the restoring force decreases sharply, leading to large-amplitude rolling and even capsizing of the ship. However, current numerical prediction methods for such motions lack sufficient accuracy to meet the actual requirements of ship safety. To further explore the mechanism of the nonlinear roll motion during pure loss of stability in following waves and improve the accuracy of numerical prediction, this study develops a numerical method based on a six-degree-of-freedom (6−DOF) weakly nonlinear time-domain model derived from the unified theory.

This proposed 6−DOF model not only effectively couples the dynamic characteristics of seakeeping and maneuvering, which provide a key advantage over traditional single-function models, but also incorporates the hydrodynamic lift force through the vortex lattice method (VLM). The introduction of VLM allows for accurate representation of the lateral fluid effects induced by variations in heel angle and ship speed, since these two factors play a crucial role in the generation of hydrodynamic lift. However, a notable limitation of VLM is that it may overestimate the lift force under certain working conditions, which will affect the reliability of the model. To address this problem, the computational fluid dynamics (CFD) method is adopted. Specifically, CFD is used to conduct a detailed and quantitative analysis of the vortex shedding behavior of the ship under self-propulsion conditions. By obtaining accurate data on the intensity and spatial distribution of vortices through CFD simulations, the lift force calculated by VLM is effectively corrected, thereby reducing the overestimation bias.

To evaluate the performance of the modified 6−DOF model, a series of comparative analyses are conducted between the predicted results of the model and the published model test data. The results indicate a clear correlation between ship speed and the effect of lift force on roll response. As ship speed increases, the lift force exerts a more significant amplification effect on the ship's roll response. When the Froude number (Fr) reaches 0.330, the amplifying effect is particularly pronounced, leading to a substantial increase in roll amplitude and a greater discrepancy between the results with and without lift force. Moreover, correcting the lift force using CFD not only reduces the overestimation of lift force by VLM but also significantly improves the predictive accuracy of the 6−DOF model for nonlinear roll motion in following waves. At Fr = 0.250, the modified model can more accurately reproduce the periodic stable roll motion, with both the phase and amplitude of the motion showing better agreement with the test data compared with those of the unmodified model.

This study makes several important contributions to the field of ship stability. First, it clarifies the influence law of the lift effect on the nonlinear roll motion of ships in following waves, revealing that the lift effect becomes more pronounced as ship speed increases, which provides a key theoretical basis for understanding the stability characteristics of ships under high-speed conditions. Second, it validates the effectiveness of the 6−DOF weakly nonlinear model based on the "VLM + CFD correction" approach in predicting ship roll motion. This model addresses the limitations of traditional models, offering higher prediction accuracy and broader applicability. Finally, the research results provide crucial technical support for ship stability assessment in engineering practice. These results can help engineers more accurately evaluate the stability performance of ships during the design phase and also provide a reliable reference for the formulation of navigation strategies under severe sea conditions, thereby enhancing the safety of ship navigation.

This study aims to calculate and analyze the capsizing probability of damaged vessels in beam seas under dead ship conditions, considering varying significant wave heights and characteristic wave periods. The analysis is conducted using the direct counting method and a dynamic model. The objective is to refine and improve the existing conservative estimation method based on residual stability parameters used in damaged vessel stability assessment regulations.

The dynamic model incorporates the coupled interactions among waves, the vessel, and internal floodwater. The wave-ship coupling is modeled by calculating nonlinear restoring forces and Froude−Krylov forces through instantaneous integration of pressures over the wetted surface. Radiation and diffraction forces are obtained using the STF method in combination with impulse response functions, while experimentally derived roll damping coefficients are included to capture viscous effects. The wave-floodwater coupling is accounted for through modified Bernoulli equations that determine inflow and outflow rates at damaged openings. The ship-floodwater coupling is modeled using the lumped mass method to simulate floodwater sloshing forces. Hydrostatic pressures induced by flooding are evaluated via instantaneous integration over the wetted surface, and viscous dissipation effects are approximated using semi-empirical formulations. The direct counting method, following the International Maritime Organization (IMO) guidelines, is employed to quantify capsizing probability using time-dependent marginal probability functions.

The proposed model is applied to the benchmark damaged vessel DTMB 5415 to evaluate roll motion responses and capsizing probabilities. The key findings are as follows: 1) The dynamic model’s predicted roll motion responses exhibit relative errors within 20% compared to experimental data near the period corresponding to the peak response. 2) The 30-minute capsizing probability increases with significant wave height, reaching its maximum when the wave characteristic period approaches the natural roll period of the damaged vessel.

The proposed methodology provides an objective and accurate approach for predicting the capsizing probability of damaged vessels in beam seas. It demonstrates significant improvements over conventional residual stability-based approaches.