Unsteady critical liquid sheet flows

Abstract

The unsteady global dynamics of a gravitational liquid sheet interacting with a one-sided adjacent air enclosure (commonly referred to as nappe oscillation configuration) is addressed under the assumptions of potential flow and the presence of surface tension effects. From a theoretical viewpoint the problem is challenging, because from previous literature it is known that the equation governing the evolution of small disturbances exhibits a singularity at the vertical station where the local flow velocity equals the capillary wave velocity (local critical condition), although the solution to the problem has not yet been found. The equation governing the local dynamics resembles one featuring the forced vibrations of a string of finite length, formulated in the reference frame moving with the flow velocity, and exhibits both slow and fast characteristic curves. From the global system perspective the nappe behaves 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 displacement of the compliant nappe centreline. A suited procedure is developed to remove the singularity of the integro-differential operator for Weber numbers less than unity. The investigation is carried out by means of a modal (i.e. time asymptotic) linear approach, which is corroborated by numerical simulations of the governing equation and supported by systematic comparisons with experimental data from the literature, available in the supercritical regime only. As regards the critical regime for the unit Weber number, the major theoretical result is a sharp increase in oscillation frequency as the flow Weber number is gradually reduced from supercritical to subcritical values due to the shift of the prevailing mode from the slow one to the fast one.