Global dynamics of gravitational liquid sheet flows

Unsteady free-interface vertical liquid sheet flows are studied from the global viewpoint, where the dynamics is termed global because it refers to the whole fluid system. The development of a proper mathematical model is presented initially, which accounts of pressure disturbances produced by the compliant interface in an air enclosure adjacent to the sheet. Our study has been restricted to the sinuous (unsymmetric) solution of the linearized set of equations. It is found that, in absence of surface tension, the optimal disturbance energy exhibits a transient growth characterized by high frequency and low frequency time-periodic oscillations; physical considerations are developed in order to estimate the relevant periods. In presence of surface tension, the low frequency oscillations disappear and the optimal disturbance energy goes quickly to zero, after exhibiting an initial reduced peak. In order to give insight on the physical relevance of such behaviours, an equation of energy budget is also derived which is used to estimate the contribution of the various physical effects evaluated via direct numerical simulation of the linearized model.