Search

It is known that the basic nature of the global dynamics of gravitational (and, therefore, spatially developing) liquid sheet flows, in the presence of surface tension effects, depends crucially1,2 on the inlet Weber number,  2 / We  Uo ho , where  is the liquid density, Uo is the inlet liquid velocity, ho is the initial half thickness of the sheet,  is the surface tension at the air-liquid interface. When We 1 , the local Weber number too is greater than unity at each streamwise (i.e., vertical) location, so that the flow can be defined supersonic everywhere, in the sense that the local flow velocity exceeds the capillary waves velocity  / h , where h is the local sheet thickness. On the other hand, when We 1 , there exists an initial region where the sheet flow is subsonic, up to the transonic location, downstream of which the flow becomes supersonic. Since the experimental evidence is that the sheet breaks-up in flow conditions of We 1 only, the analysis of the flow field initially subsonic, and hence including the transonic line, is mandatory to predict the rupture conditions. From the theoretical viewpoint the problem is not straightforward, because the equation governing the evolution of global linear disturbances exhibits a singularity just at the transonic station, as documented also by previous contributions of literature3,4, although the solution to the problem is not yet yielded. Thus, the present work is aimed at developing a theoretical/numerical procedure to fill up this lack of information. Within this framework, the flow is assumed inviscid and the problem is arranged in 1D formulation along the streamwise direction by expressing all the physical quantities through a coordinate-type expansion in terms of powers of the local lateral distance from the centerline position. The interaction with the external environment refers to an air enclosure located on one side of the curtain. The linearized perturbation equations are determined in a standard fashion by superimposing infinitesimal disturbances to the steady solution and the modal global stability is studied by addressing the relevant singular eigenvalues problem. The dimensionless governing equation of sinuous disturbances can be written as:    where U  1 2x is the classic free-fall Torricelli’s solution, f is the centerline deflection of the sheet, k is a proper compressibility coefficient of the air enclosure and L is the curtain length. It is shown that the transonic location can be classified as a regular singular point and a solution in terms of Frobenius series5 is introduced in order to provide the local solution around the singularity. The non-linear eigenvalues problem obtained accordingly is faced as a nonlinear two-point boundary problem (in which the searched eigenvalue is considered as an unknown function) that is solved by means of a shooting technique6. The boundary conditions are of null sheet displacement at inlet and boundedness at the singularity location3,4. The numerical results show the spectra pattern obtained when the inlet We number is varied from subsonic to transonic values, with particular attention to their evolution with respect to cases of entirely supersonic flow analyzed with standard spectral methods. The physical relevance of the present findings is discussed as well. a University of Naples Federico II, Dept. Industrial Eng., Aerospace Sector, Piazzale Tecchio 80, Naples, ITALY, deluca@unina.it 1 de Luca, J. Fluid Mech. 399, 355 (1999). 2 Le Grand-Piteira et al., Phys. Rev. E 74, 026305 (2006) 3 Finnicum et al., J. Fluid Mech. 255, 647 (1993). 4 Weinstein et al., Phys. Fluids 9, 1815 (1997). 5 Bender and Orszag, Springer (1999). 6 Press et al., Cambridge University Press (1992). 441

The unsteady global dynamics of a gravitational liquid sheet interacting with a onesided adjacent air enclosure, typically referred to as nappe oscillation, is addressed, under the assumptions of potential flow and absence of surface tension effects. To the purpose of shedding physical insights, the investigation examines both the dynamics and the energy aspects. An interesting re-formulation of the problem consists of recasting the nappe global behavior 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 average displacement of the compliant nappe centerline. The investigation is carried out through a modal (i.e., time asymptotic) and a non-modal (i.e., short-time transient) linear approach, which are corroborated by direct numerical simulations of the governing equation. The modal analysis shows that the flow system is characterized by low-frequency and high-frequency oscillations, the former related to the crossing time of the perturbations over the whole domain and the latter related to the spring-mass oscillator. The low-frequency oscillations, observed in real life systems, are produced by the (linear) combination of multiple modes. The non-normality of the operator is responsible for short-time energy amplifications even in asymptotically stable configurations, which are confirmed by numerical simulations and justified by energy budget considerations. Strong analogies with the edge-tone problem are encountered; in particular, the integer-plus-one-quarter resonance criterion is uncovered, where the basic frequency to be multiplied by n + 1/4 is just the one related to the spacing among the imaginary parts of the eigenvalues.