Resonant three-dimensional nonlinear sloshing in a square base basin. Part 5. Three-dimensional non-parametric tank forcing

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, 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 et al., J. Fluid Mech., vol. 487, 2003, pp. 1-42). Constructing and analysing asymptotic periodic solutions of the system made it possible to classify steady-state 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, but no heave) cyclic tank motions. It becomes possible after establishing an asymptotic equivalence of the associated periodic solutions of the modal system to 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-axis ratio and direction (counter- or clockwise) of the equivalent orbits. 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 response curves for wall-symmetric (canonic) and diagonal orbit positions. This is not true for the oblique-type elliptic forcing. When the semi-axis ratio changes from 0 to 1, the response curves exhibit astonishing metamorphoses significantly influencing the frequency ranges of stable nearly standing/swirling waves and 'irregular' sloshing. For the experimental input data by Ikeda et al. (J. Fluid Mech., vol. 700, 2012, pp. 304-328), the counter-directed swirling disappears as but the frequency range of irregular waves vanishes for 0.75 less than or similar to vertical bar delta(1)vertical bar.