Damping of the swirling wave mode

We generalize recent results of the authors [A. Miliaev and A. Timokha, Viscous damping of steady-state resonant sloshing in a clean rectangular tank, J. Fluid Mech., 965, № R1, 1 – 11 (2023); DOI: doi.org/10.1017/jfm.2023.372] on machine learning of multidimensional modal systems derived to describe nonlinear sloshing in rigid tanks exposed to resonant excitations. The multimodal systems neglect viscous damping but the learning procedure pursues incorporating a priori unknown damping terms into them. The main focus is on steady-state swirling wave regime in an upright circular tank which moves longitudinally by a harmonic law with the excitation frequency close to the lowest sloshing eigenfrequency. The nonlinear Narimanov – Moiseev-type modal system is employed with a set of phenomenological linear and nonlinear damping quantities whose coefficients are a subject of the learning procedure. The procedure requires semi-analytical periodic (steady-state) solutions of the Narimanov – Moiseev-type system which are constructed in the present paper. Based on these solutions and the loss (cost) function expressing the "distance" between measured and theoretical phase-lags for the swirling wave regime, the constructed learning procedure is tested with experimental data by A. Royon-Lebeaud, E. J. Hopfinger, and A. Cartellier [Liquid sloshing and wave breaking in circular and square-base cylindrical containers, J. Fluid Mech., 577, 467 – 494 (2007); DOI: doi.org/10.1017/S0022112007004764].