Resonant three-dimensional nonlinear sloshing in a square base basin. Part 5. Three-dimensional non-parametric tank forcing
Peer reviewed, Journal article
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OriginalversjonJournal of Fluid Mechanics. 2020, 894 . 10.1017/jfm.2020.253
Assuming an inviscid incompressible liquid (with irrotational flows) partly filling a square base tank, which performs a small-amplitude sway/surge/pitch/roll periodic motion whose frequency is close to the lowest natural sloshing frequency and implementing the multimodal method, a nine-dimensional Narimanov-Moiseev–type (modal) system of ordinary differential equations with respect to the hydrodynamic generalised coordinates was derived in the part 1 (Faltinsen, Rognebakke & Timokha 2003). Constructing and analysing asymptotic periodic solutions of the system made it possible to classify steadystate resonant sloshing and its stability for the harmonic reciprocating (longitudinal, diagonal and oblique) forcing. The results were supported by experimental observations and measurements. The present paper finalises the case studies by considering the three-dimensional non-parametric (combined sway, pitch, surge, roll and yaw) cyclic tank motions. It becomes possible after establishing an asymptotic equivalence of the associated periodic solutions of the modal system and those for a suitable horizontal translatory elliptic forcing so that, as a consequence, resonant steady-state waves and their stability can be considered versus angular position, semi-axes ratio |δ1| and direction (counter- or clockwise) of the equivalent orbit. The circular orbit causes stable swirling waves (co-directed with the orbit) but may also excite stable nearly-standing waves. The orbit direction does not affect the wave-amplitude response curves for wall-symmetric (canonic) and diagonal angular positions. The latter is not true for the oblique-type elliptic forcing. When the semi-axes ratio |δ1| changes from 0 to 1, the response curves exhibit astonishing metamorphoses significantly influencing the frequency ranges of stable nearly-standing/swirling wave modes and irregular (chaotic) sloshing. For the experimental input data by Ikeda et al. (2012), the counter-directed swirling disappears as 0.5 . |δ1| but the frequency range of irregular waves vanishes for 0.75 . |δ1|.