Study on simplified calculation method for lateral natural frequency of water within U-shaped aqueducts.

This article focuses on developing a simplified and efficient calculation method for the lateral natural frequencies of water sloshing within two-dimensional (2D) U-shaped aqueducts, which are widely used in water diversion projects in China. Building on an existing Rayleigh quotient-based approach for arbitrary-shaped 2D containers, the method employs Taylor series expansions and the beta (β) and gamma (Γ) functions to transform complex integrals into algebraic expressions, enabling precise calculation of the first four lateral natural frequencies. Validation through numerical simulations using OpenFOAM-9 demonstrates that this method is approximately nine times more accurate than the current Chinese seismic design code for the first lateral natural frequency and can reliably compute higher-order frequencies. The study also finds that when the water level exceeds the semicircular part of the aqueduct, the aqueduct width significantly affects the natural frequencies, whereas water depth has minimal influence. The proposed method offers a practical tool for engineers and is recommended for adoption in design codes, though further experimental validation is suggested.