Nonlinear interactions and energy exchange between liquid sloshing modes

Non-stationary effects of the resonance energy exchange between liquid sloshing modes for square-base tanks are analyzed in terms of new descriptive variables namely energy partition and coherency (phase shift) indexes. In particular, it is shown that such a couple represents conjugate variables of an effective Hamiltonian system whose phase plane captures all possible effects of the modal interaction. Furthermore, the presence of damping affects only the temporal scale of the dynamics but still preserves the Hamiltonian structure of equations that provides the existence of first integral. As a result, analytical solution for the nonlinear interaction of predominant modes is obtained in quadratures. The dynamic properties of the modal interaction are controlled by a single parameter which depends upon the fluid depth. Transitions of the phase plane diagram reveal that, above some critical depth, the diagonal in-phase and out-of-phase sloshing modes disappear while only clockwise and counterclockwise swirling, and running phase modes are possible. Whereas the quadratures are invertible within the class of elliptic functions, an explicit solution for the important critical case is obtained in terms of elementary functions. The approach is illustrated on the sloshing model for a square-base tank suggested by Ikeda et al. (T. Ikeda, R.A. Ibrahim, Y. Harata, T. Kuriyama, Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation, J. Fluid Mech., 700 (2012) 304-328.) However, formulated in general terms of coupled oscillators, the approach seems to have a wider area of applicability dealing with non-stationary effects of resonance interactions with nonlinear beating effects in physics and classical mechanics.