The unsteady dynamics of planar liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is numerically investigated by means of the Volume-of-Fluid (VOF) technique for supercritical regimes. The global behavior of the non-parallel flow is analyzed by perturbing the initial steady configuration by means of a Gaussian bump in the transverse velocity component of relatively small amplitude, thereby exciting sinuous modes. To gain more physical insights into the fluid system, a theoretical linear one-dimensional model is also developed. A physical interpretation of this model relates the sheet dynamics to transverse vibrations of tensional string forced by terms containing the lateral velocity and subjected to a total damping coefficient, which can assume negative values. The VOF simulation satisfactorily confirms that the velocity impulse perturbation splits into two wave fronts traveling downstream with the theoretical wave velocities. A good agreement is found in comparing the crossing times over the entire domain length of such waves with the almost constant spacing between the frequencies of the eigenvalue spectrum. Surface tension plays a stabilizing role, and for relatively high values of density ratio rρ of gaseous-to-liquid phases, the sheet becomes unstable. It is argued that the distribution of transverse velocity component of the gaseous phase represents the forcing term, which leads the system toward the instability when, for relatively high rρ, the total damping becomes negative. An analogy seems to exist between the global unstable behavior exhibited by the liquid sheet as rρ increases and the shear-induced global instability found by Tammisola et al. [Surface tension-induced global instability of planar jets and wakes,” J. Fluid Mech. 713, 632–658 (2012)] in the presence of surface tension. However, for the gravitational sheet, the surface tension is stabilizing.

The receptivity to forcing harmonic disturbances of transverse velocity in subcritical liquid sheet flows subjected to gravity is studied. The investigation is carried out both by employing the linear stability theory applied to a simplified one-dimensional inviscid model and by performing fully two-dimensional numerical simulations based on the Volume-of-Fluid technique. The computation of global sinuous eigenmodes and eigenvalues has required the removal of the singularity of the governing equation, for the first time carried out in the case of unconfined gaseous ambient. Direct numerical simulations of the unsteady sheet when continuously forced by a perturbation in lateral velocity are reported. The harmonic forcing, applied at the inlet section, basically excites sinuous modes of the system, related to the natural impulse response. The results of receptivity have been treated by employing a proper one-dimensional reduction technique to compare numerical data with the corresponding findings of the stability theory. Depending on the Reynolds number, two different behaviors are observed: at low Re the large viscous effect makes the system overdamped; as Re increases and the inviscid conditions are approaching, the frequency response exhibits a peak frequency (resonance) which closely agrees with the frequency of the least stable eigenvalue. The various stations synchronize with the critical station as Re increases, and therefore it forces the global oscillations of the flow field. This behavior of the critical station retrieves the role of wavemaker, which fails for high-frequency forcing. The resonance characteristics of the sheet have been further analyzed by inspecting the fully two-dimensional velocity fields. A major finding at low forcing frequency is the nonlinear varicose distortion of the sheet thickness that progressively envelops the basic sinuous shape when the inviscid conditions are approaching.

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 $We$ 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.

A data-driven approach to estimate the global spectrum of gravitational planar liquid jets (sheet or curtain flows) is presented in this work. The investigation is carried out by means of two-dimensional numerical simulations performed through the solver BASILISK, based on the one-fluid formulation and the volume-of-fluid approach. The dynamic mode decomposition technique is applied to extract the underlying linear operator, considering random perturbations of the base flow. The effectiveness of this procedure is first evaluated comparing results with those of a simplified one-dimensional curtain model in terms of spectrum and eigenfunctions. The methodology is then applied to a two-dimensional configuration obtaining the BiGlobal spectra for both supercritical (Weber number We > 1) and subcritical ( We < 1) regimes. Results highlight that in supercritical regime, the spectrum presents three branches: the upper and lower ones exhibit a purely sinuous behavior with frequencies quite close to those predicted by the one-dimensional model; the middle branch presents a predominant varicose component, increasing with the frequency. The subcritical spectrum, instead, shows that the first two less stable eigenvalues, sorted by increasing frequency, exhibit, respectively, a sinuous and a varicose behavior, while their growth rate is almost the same. As expected, the subcritical regime does not reveal the slow branch. The effect of the density ratio, [Formula: see text], between the two phases is investigated, revealing that the flow system is unstable for [Formula: see text]. Topological inspections of the leading modes in this unstable configuration show that the predominance of a varicose behavior is related to the rupture of the curtain.

The unsteady dynamics of a gravitational liquid sheet, driven by a continuous harmonic perturbation in the lateral velocity component applied at the inlet section, is analyzed. The topology and the dynamics of the relevant flow structures are characterized by applying POD (Proper Orthogonal Decomposition) and spectral POD (SPOD) modal decompositions on two-dimensional two-phase numerical simulation data obtained with the volume-of-fluid approach. The investigation is carried out by varying the Weber number, the forcing frequency (Strouhal number), and the Reynolds number. The supercritical regime (We &gt; 1) features a traveling perturbation, exhibiting a spatial structure with leading sinuous modes. SPOD spectra confirm the occurrence of a discontinuity in frequency response between the supercritical and subcritical regimes. In the subcritical regime (We &lt; 1), the investigation highlights the excitation of a combined sinuous–varicose motion when the system is driven at resonance frequency for a relatively high Reynolds number (approaching the inviscid limit). The emergence of varicose modes is favored by low Weber numbers. The excitation of these modes occurs when the Weber number is decreased from We = 0.90 down to 0.75, with a progressive shift of the varicose mode from higher harmonics toward the main frequency; it can be considered as a possible mechanism of breakup observed in experiments when the inlet flow rate is progressively reduced. The flow reconstruction based on both POD and SPOD confirms the good capability of SPOD modes to capture dynamically relevant features of the fluid motion in subcritical conditions.

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An energy budget approach based on numerical simulations of a linear low-order model, combined with linear global stability analysis, is used to investigate the unsteady dynamics of subcritical ( We <  1) gravitational liquid sheet flows. It is found that surface tension is the physical mechanism responsible for the modal flow instability as the Weber number is progressively decreased down to a critical threshold Weth for which the sheet is entirely subcritical. A transient algebraic growth of the perturbation characterized by the power law [Formula: see text] is found in both asymptotically stable ([Formula: see text]) and unstable ([Formula: see text]) conditions. This finding agrees with a previous result of the literature obtained by employing a local spatiotemporal stability technique (for an infinite domain) for which in the subcritical regime an absolute instability occurs. However, in the present study, the temporal evolution of disturbances in the unstable case eventually follows an asymptotic modal growth, which is also recovered in the eigenvalue spectra evaluated using linear stability analysis. Asymptotic stability of the flow detected in the range [Formula: see text] is not caused by the damping effect of viscosity, but by the energy exchanges through the domain boundaries. Surface tension-induced instability is further studied by means of parametric analysis involving the Froude number Fr and the slenderness ratio parameter ε. It is found that decreasing ε and increasing Fr have the same destabilizing effect. The present work represents a further step toward a deeper understanding of liquid sheet dynamics in the subcritical regime, with the aim of providing a theoretical background to establish connections between results of two-dimensional modeling and three-dimensional observations of real occurrence.

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