Global dynamics analysis of nappe oscillation

The unsteady global dynamics of a gravitational liquid sheet interacting with a onesided adjacent air enclosure, typically referred to as nappe oscillation, is addressed, under the assumptions of potential flow and absence of surface tension effects. To the purpose of shedding physical insights, the investigation examines both the dynamics and the energy aspects. An interesting re-formulation of the problem consists of recasting the nappe global behavior as a driven damped spring-mass oscillator, where the inertial effects are linked to the liquid sheet mass and the spring is represented by the equivalent stiffness of the air enclosure acting on the average displacement of the compliant nappe centerline. The investigation is carried out through a modal (i.e., time asymptotic) and a non-modal (i.e., short-time transient) linear approach, which are corroborated by direct numerical simulations of the governing equation. The modal analysis shows that the flow system is characterized by low-frequency and high-frequency oscillations, the former related to the crossing time of the perturbations over the whole domain and the latter related to the spring-mass oscillator. The low-frequency oscillations, observed in real life systems, are produced by the (linear) combination of multiple modes. The non-normality of the operator is responsible for short-time energy amplifications even in asymptotically stable configurations, which are confirmed by numerical simulations and justified by energy budget considerations. Strong analogies with the edge-tone problem are encountered; in particular, the integer-plus-one-quarter resonance criterion is uncovered, where the basic frequency to be multiplied by n + 1/4 is just the one related to the spacing among the imaginary parts of the eigenvalues.