Modal analysis of three-dimensional gravitational thin liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is performed in the present work. Numerical data of this relevant two-phase flow configuration are obtained through the single-phase formulation and the Volume-of-Fluid (VOF) technique implemented in the flow solver Basilisk [1]. This class of flows exhibits a variety of spatially and dynamically relevant structures, both in free and forced configurations, that are investigated through modal decomposition techniques, such as Proper Orthogonal Decomposition (POD), Spectral Proper Orthogonal Decomposition (SPOD) and Dynamic Mode Decomposition (DMD) [2]. Moreover, we employ also autoencoders to achieve the dimensionality reduction. By means of these methodologies, we explore the effect of two main governing parameters on the flow dynamics, namely the liquid sheet aspect ratio, AR = W/H, where H and W are the sheet inlet thickness and width, and the Weber number, W e = ρ l U H/(2σ), in which U is the inlet liquid velocity, ρ l the liquid density, and σ the surface tension coefficient. Finally, for the highest aspect ratio value considered (AR = 40), we investigate the forced dynamics of the system excited by a harmonic perturbation in transverse velocity component applied at the inlet section, comparing results with ones arising from a purely two-dimensional analysis of the flow [3]. The obtained results highlight the low rank behavior exhibited by the flow, suggesting that reduced order modeling could be particularly appealing to reduce complexity and computational effort in numerical simulation of this class of flows.

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It is known that the disintegration of vertical liquid curtains (sheets) is affected crucially by the amplification of free edge holes forming inside the curtain. This paper aims to investigate the influence of the hole expansion dynamics, driven by the so-called rim retraction, on the breakup of a liquid curtain, in both supercritical (Weber number $We > 1$ ) and subcritical ( $We < 1$ ) conditions. The analysis is based on three-dimensional direct numerical simulations. For a selected supercritical configuration, the steady flow topology is first analysed. The investigation reveals the classic triangular shape regime of the steady curtain, due to the surface-tension-induced borders retraction towards its centre plane. The unsteady dynamics is then investigated as the curtain response to a hole perturbation introduced artificially in the steady flow configuration. The hole evolution determines a rim retraction phenomenon inside the curtain, which is influenced by both capillary and gravity forces. In supercritical conditions, the hole does not influence the curtain flow dynamics in the long-time limit. By reducing the Weber number slightly under the critical threshold ( $We=1$ ), the initial amplification rate of the hole area increases, due to the stronger retraction effect of surface tension acting on the hole rims. The free hole expansion in fully subcritical conditions ( $We < 1$ ) is investigated finally by simulating an edge-free curtain flow. As $We$ decreases progressively, the hole expands while it is convected downstream by gravity acceleration. In the range $0.4< We<1$ , the subcritical curtain returns to the intact unperturbed configuration after the hole expulsion at the downstream outflow. For $We<0.4$ , the surface tension force becomes strong enough to reverse the gravitational motion of the hole top point, which retracts upstream towards the sheet inlet section while expanding along the lateral directions. This last phenomenon causes finally the breakup of the curtain, which results in a columnar regime strictly resembling similar experimental findings of the literature.