Multimodal analysis of weakly nonlinear sloshing in a spherical tank

Sloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963, Tech. Rep. TN D-1991, NASA Technical Note) and Sumner (1966, Tech. Rep. TN D-3210, NASA). A good agreement is established for $0. 2\leq h\lesssim 1$ . Discrepancy for higher liquid depths $1\lesssim h\lt 2$ are explained by secondary resonance.