Objectives Parametric roll is a typical stability-loss mode considered in the International Maritime Organization (IMO) second-generation intact stability framework and is most frequently reported 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 realistic operations, however, ships often sail in bow oblique seas, where the roll response may be governed by the combined effects of parametric excitation and direct wave-forced excitation, particularly 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 remain less systematically understood. Accordingly, this study aims to (i) characterize the parametric roll behavior of a ship advancing in bow oblique waves, (ii) quantify the relative roles 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 regulates the transition between excitation mechanisms, thereby providing guidance for stability improvement in oblique-wave environments.

Methods 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 represent wave nonlinearity relevant to practical sea states. The study comprised two stages. (1) Validation: simulations were conducted for KCS parametric roll in head seas using publicly available constrained model-test data, and key roll metrics (amplitude and period) were compared to 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 impacts on roll dynamics and excitation characteristics. The simulations employed a 1:100 geometrically scaled simplified KCS model with rudder; the heave, roll, and pitch degrees of freedom were released. A cylindrical computational domain was used and discretized 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, including grid-spacing uncertainty U_G and time-step uncertainty U_T.

Results The validation results demonstrate that the numerical framework reproduces the head-sea parametric roll response with high fidelity: the mean deviation of roll amplitude and period across different forward speeds is within 5%, and the maximum relative error is below 10%. In bow oblique waves, the KCS exhibits simultaneous responses associated with 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., 0.5f_e, 1.5f_e, ...) and harmonics associated with wave-forced excitation (e.g., f_e, 2f_e, ...), where f_e denotes the encounter frequency. Two parametric dependencies are highlighted. (1) Wave heading angle: increasing the heading angle progressively weakens parametric excitation while enhancing wave-forced excitation. The roll amplitude remains nearly unchanged for heading angles between 0°–36°, but decreases markedly in the 36°–60° range, 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 are no longer observed. (2) Wave steepness: increasing wave steepness strengthens the wave-forced contribution and induces a nonlinear modification of the restoring moment consistent with a hardening-type behavior, leading to an asymptotic reduction of parametric-roll amplitude. When H/λ=0.06 (with H the wave height and λ the wavelength), the maximum roll angle decreases to about 5°, indicating substantial mitigation of roll severity. Moreover, a larger wave heading angle enhances wave-forced excitation while attenuating the steepness-induced hardening effect.

Conclusions The study confirms that the HUST-SWENSE-based numerical framework can accurately predict parametric roll 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 regulate ship parametric roll in bow oblique seas and reveal the mechanism underlying 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 for 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., via bilge keels or other anti-roll devices) to suppress resonance amplitude and improve stability margins.