Abstract
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 .
Highlights
Suggesting an inviscid liquid with irrotational flows, a finite liquid depth, a forcing frequency σ close to the lowest natural sloshing frequency σ1, a forcing amplitude asymptotically smaller than the tank width/breadth (the five non-dimensional sway/surge/roll/pitch/yaw amplitudes are O() 1) and assuming that secondary resonance phenomena can be neglected, Faltinsen, Rognebakke & Timokha (2003, Part 1) derived a Narimanov–Moiseev-type nonlinear modal system, which effectively approximates resonant sloshing in a square base tank
Three generalised coordinates have the second asymptotic order, O( 2/3 ), but the other four have the third asymptotic order O(). This number of hydrodynamic generalised coordinates and their asymptotic order on the O()-scale are a consequence of classical mathematical results by Moiseev (1958) and Narimanov
A cyclically moving rigid square base container is partially filled with a finite liquid depth by a perfect incompressible liquid
Summary
Suggesting an inviscid liquid with irrotational flows, a finite liquid depth, a forcing frequency σ close to the lowest natural sloshing frequency σ1 , a forcing amplitude asymptotically smaller than the tank width/breadth (the five non-dimensional sway/surge/roll/pitch/yaw amplitudes are O() 1) and assuming that secondary resonance phenomena can be neglected, Faltinsen, Rognebakke & Timokha (2003, Part 1) derived a Narimanov–Moiseev-type nonlinear modal system (of ordinary differential equations), which effectively approximates resonant sloshing in a square base tank. Because the first- and second-order components of the constructed asymptotic periodic solution and its stability are functions of a, ā, b, b, Λ, and, are functions of x , ̄x , ̄y , y (= functions of the aforementioned lowest Fourier harmonics), the original stable/unstable theoretical steady-state waves due to the complex periodic sway, surge, roll, pitch and yaw become asymptotically (to within the higher-order contribution) equivalent to the steady-state sloshing due to the aforementioned horizontal elliptic tank motions Based on this fact, the steady-state wave modes can be classified versus 0 6 |δ1 | 6 1. The present paper should guide future experimental studies with a focus on co-existing stable resonant waves established for all forcing types
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