Natural frequency discontinuity of vertical liquid sheet flows at transcritical threshold

The natural and forced dynamic response of a gravitational plane liquid sheet (curtain) of finite length interacting with an unconfined gaseous ambient is numerically and experimentally investigated. The global eigenvalue spectrum obtained by means of a linear inviscid one-dimensional model, accounting for the coupling between the curtain motion and the ambient pressure disturbances, clearly shows an abrupt increase (jump) in the characteristic natural frequency of the flow when the supercritical (We>1) to subcritical (We<1) transition occurs, with the Weber number defined as the ratio between inertia and capillary forces. On the other hand, the numerical simulation of the forced sheet response does not show any discontinuity between supercritical and subcritical conditions, as recently found by Torsey et al. (J. Fluid Mech., vol. 910, 2021, pp. 1-14) in the case of an infinite liquid sheet subjected to imposed ambient pressure disturbances not coupled with the curtain motion. It is argued that the forced liquid sheet behaviour varies continuously in shape and amplitude between the two regimes, not depending on the specific liquid-gas interaction model considered, whilst the natural frequency of the finite flow system does undergo a discontinuity, which can be theoretically predicted by the model of sheet-ambient interaction employed here. As a major result, the experimental evidence of the natural frequency jump is for the first time provided as well.