Simplified semi–analytical prediction of longitudinal sloshing natural periods in head-ended type-C tanks under cryogenic and liquid-hydrogen-relevant conditions

The natural period of sloshing in independent Type-C tanks is a key parameter for resonance avoidance, test-condition selection, and computational fluid dynamics (CFD)-based load assessment. Commonly used DNV-type formulas, however, do not account for two effects specific to head-ended cryogenic tanks: (i) the filling-level-dependent effective free-surface length introduced by the ellipsoidal heads, and (ii) the non-negligible ullage-gas density that modifies the interfacial restoring force. This study develops a simplified semi-analytical framework for the first two longitudinal sloshing periods in head-ended Type-C tanks, combining a filling-dependent effective length, derived in closed form from the head geometry, with a density-ratio correction introduced through an effective gravity, while preserving the closed-form structure of the guideline expression. A three-dimensional Steklov eigenvalue solution is adopted as the primary absolute-period reference, and a two-fluid gravity–capillary model serves as an auxiliary linear benchmark to separate density-ratio, finite-ullage, and surface-tension effects. For a 2.5 m³-class head-ended Type-C tank, the head correction shortens the predicted period by up to about 10% at extreme fillings, whereas the density-ratio and finite-ullage effects increase the predicted period by up to about 2% and 4%, respectively, for the representative near-atmospheric liquid hydrogen (LH 2 )-like properties. Using the experimental P10 dataset of Park (2026), obtained under DNV-type excitation, an illustrative detuning-oriented consistency check shows a strong negative correlation between the predicted detuning ratio and the normalized peak impact pressure; because those tests were excited at the baseline period, this is interpreted as physical evidence of detuning-induced load reduction rather than as a direct absolute-period validation, for which the 3-D Steklov reference is used. The resulting expression is intended as a transparent engineering screening tool that corrects the dominant first-order shifts associated with effective free-surface length and gas-liquid density ratio, suited to resonance screening, test planning, and preliminary CFD setup rather than to direct nonlinear impact-pressure prediction.