Parametric roll is a common stability loss mode considered in the international maritime organization (IMO) second-generation intact stability framework, most often occurring in head or following seas. It manifests as large-amplitude roll motions coupled with pronounced heave–pitch responses, posing risks to personnel safety and potentially causing structural damage to the ship and cargo. In practical operations, however, ships frequently operate in bow oblique seas, where the roll response is influenced by the combined effects of parametric excitation and direct wave-forced excitation, especially when the wave encounter frequency approaches approximately twice the ship's natural roll frequency. Compared with head-sea conditions, the coexistence and interaction of multiple excitation mechanisms in bow oblique seas are still not fully understood. Therefore, this study aims to (i) characterize the parametric roll behavior of a ship advancing in bow oblique waves, (ii) quantify the relative contributions of parametric and wave-forced excitations by analyzing the time-varying and spectral features of the restoring (righting) moment, and (iii) elucidate how the wave heading (wave direction) angle influences the transition between excitation mechanisms, thereby providing guidance for stability improvement in oblique-wave environments.
Numerical simulations were performed for the KCS (KRISO container ship) advancing in bow oblique regular waves using two solvers developed at Huazhong University of Science and Technology (HUST): (1) HUST-SWENSE, a functional-decomposition-based potential–viscous flow coupled solver, and (2) HUST-Overset, a structured dynamic overset-grid solver. A fifth-order Stokes nonlinear regular-wave model was incorporated to capture wave nonlinearity relevant to practical sea states. The study comprised two stages. (1) Validation: simulations of KCS parametric roll in head seas were performed using publicly available constrained model-test data. Key roll metrics (amplitude and period) were compared with measurements to verify the numerical framework. (2) Bow oblique-wave simulations: a set of cases was designed by varying the wave heading angle and wave steepness to evaluate their effects on roll dynamics and excitation characteristics. The simulations employed a 1:100 geometrically scaled simplified KCS model with rudder, releasing the heave, roll, and pitch degrees of freedom. A cylindrical computational domain was employed, subdivided into background, hull, stern, and rudder sub-grids with fully structured meshes. Numerical uncertainty was assessed using the safety-factor method of Xing and Stern, accounting for grid-spacing uncertainty UG and time-step uncertainty UT.
The validation results demonstrate that the numerical framework accurately reproduces the head-sea parametric roll response, with a mean deviation of roll amplitude and period within 5% across different forward speeds, and a maximum relative error below 10%. In bow oblique waves, the KCS exhibits simultaneous responses arising from parametric excitation and wave-forced excitation within specific ranges of wave steepness and frequency. Spectral analysis indicates that the roll response and restoring moment contain subharmonic components related to parametric excitation (e.g., 0.5fe, 1.5fe, etc.) and harmonic components associated with wave-forced excitation (e.g., fe, 2fe, etc.), where fe denotes the encounter frequency. Two parametric dependencies are highlighted. (i) Wave heading angle: increasing the heading angle progressively weakens parametric excitation while enhancing wave-forced excitation. The roll amplitude remains nearly constant for heading angles between 0° and 36°, but decreases sharply between 36° and 60°, with an overall reduction of 83.9%. A critical heading angle of approximately 60° is identified, beyond which wave-forced excitation dominates and distinct parametric-roll features disappear. (ii) Wave steepness: increasing wave steepness amplifies the wave-forced contribution and induces a nonlinear modification of the restoring moment consistent with hardening-type behavior, leading to an asymptotic reduction in parametric-roll amplitude. When H/λ = 0.06 (where H is wave height and λ is wavelength), the maximum roll angle decreases to about 5°, indicating significant mitigation of roll severity. Moreover, a larger wave heading angle enhances wave-forced excitation while diminishing the steepness-induced hardening effect.
The study confirms that the HUST−SWENSE-based numerical framework can accurately predict the parametric roll behavior of the KCS and is suitable for analyzing roll stability in complex oblique-wave conditions. The results clarify how wave heading angle and wave steepness influence ship parametric roll in bow oblique seas, revealing the mechanism behind the transition from parametric-excitation-dominated to wave-forced-excitation-dominated responses as the heading angle increases. These findings provide a basis for numerical prediction, risk assessment, and speed/course optimization to mitigate parametric roll in oblique-wave environments. From an engineering perspective, practical mitigation measures include (i) adjusting course to avoid heading/speed combinations prone to parametric resonance and (ii) enhancing roll damping (e.g., through bilge keels or other anti-roll devices) to suppress resonance amplitude and improve stability margins.
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