Energy insights into the unsteady dynamics of a viscous gravitational liquid sheet

The impulse response of planar liquid sheet flows, subjected to gravity, and interacting with unconfined gaseous environments located on both sides of the liquid phase, is numerically investigated from an energy perspective by means of a combined approach of linear stability analysis and direct numerical simulations, carried out with the volume-of-fluid technique. The computation of global eigenmodes and eigenvalues is based on a simplified one-dimensional model also accounting for viscous effects. Physical insights are gained by means of an original energy balance equation for sinuous perturbations, identifying the energy budgets associated with the different terms governing the flow dynamics. Two distinctive features of the sheet flow, the flow instability in the supercritical regime at relatively high gas-to-liquid density ratio and the discontinuity in frequency at the supercritical-to-subcritical transition, have been recovered and discussed. The pressure work is responsible for the instability of supercritical regimes at relatively high density ratio. This finding is confirmed by the direct numerical simulations, showing a convective amplification of the perturbation as it travels downstream: for high density ratios, the large convective amplification cannot be expelled from the domain and the flow suffers from a global instability. The frequency discontinuity occurring at the supercritical-to-subcritical transition is basically due to the left-going wave expulsion; therefore, the subcritical sheet stabilizes more rapidly than the supercritical one, and the slow branch of the spectrum disappears. The high frequency oscillations observed in subcritical regime are attributed to the removal of constraint on the meanline slope when We < 1, which produces an increase in the oscillation frequency of the sheet analogous to that occurring for elastic solid beams when the clamped constraint is substituted by pinned constraint.