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

Global dynamics of transonic gravitational liquid sheet flows / Girfoglio, Michele; Fortunato De, Rosa; Coppola, Gennaro; DE LUCA, Luigi. - (2014). (Intervento presentato al convegno EFMC10 – European Fluid Mechanics Conference 10 tenutosi a Lyngby, Copenhagen nel 15-18 september 2014).

### Global dynamics of transonic gravitational liquid sheet flows

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*GIRFOGLIO, MICHELE;COPPOLA, GENNARO;DE LUCA, LUIGI*

##### 2014

#### Abstract

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). 441I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.