Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-18T14:12:08.429Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteady critical liquid sheet flows

Published online by Cambridge University Press:  18 May 2017

M. Girfoglio ,
F. De Rosa ,
G. Coppola  and
L. de Luca Open the ORCID record for L. de Luca [Opens in a new window]
M. Girfoglio
Affiliation:
Department of Industrial Engineering, Aerospace Sector, Università di Napoli ‘Federico II’, Naples, Italy
F. De Rosa
Affiliation:
Department of Industrial Engineering, Aerospace Sector, Università di Napoli ‘Federico II’, Naples, Italy
G. Coppola
Affiliation:
Department of Industrial Engineering, Aerospace Sector, Università di Napoli ‘Federico II’, Naples, Italy
L. de Luca*
Affiliation:
Department of Industrial Engineering, Aerospace Sector, Università di Napoli ‘Federico II’, Naples, Italy
*
†Email address for correspondence: deluca@unina.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The unsteady global dynamics of a gravitational liquid sheet interacting with a one-sided adjacent air enclosure (commonly referred to as nappe oscillation configuration) is addressed under the assumptions of potential flow and the presence of surface tension effects. From a theoretical viewpoint the problem is challenging, because from previous literature it is known that the equation governing the evolution of small disturbances exhibits a singularity at the vertical station where the local flow velocity equals the capillary wave velocity (local critical condition), although the solution to the problem has not yet been found. The equation governing the local dynamics resembles one featuring the forced vibrations of a string of finite length, formulated in the reference frame moving with the flow velocity, and exhibits both slow and fast characteristic curves. From the global system perspective the nappe behaves 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 displacement of the compliant nappe centreline. A suited procedure is developed to remove the singularity of the integro-differential operator for Weber numbers less than unity. The investigation is carried out by means of a modal (i.e. time asymptotic) linear approach, which is corroborated by numerical simulations of the governing equation and supported by systematic comparisons with experimental data from the literature, available in the supercritical regime only. As regards the critical regime for the unit Weber number, the major theoretical result is a sharp increase in oscillation frequency as the flow Weber number is gradually reduced from supercritical to subcritical values due to the shift of the prevailing mode from the slow one to the fast one.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Interfacial Flows (free surface) Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 219 - 247
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, N. S., Helenbrook, B. T. & Lin, S. P. 2011 Transience to instability in a liquid sheet. J. Fluid Mech. 666, 358–390.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer.CrossRefGoogle Scholar
Binnie, A. M. 1974 Resonating waterfalls. Proc. R. Soc. Lond. A 339, 435–449.Google Scholar
Boyce, W. E. & DiPrima, R. C. 2008 Elementary Differential Equations and Boundary Value Problems. Wiley.Google Scholar
Brunet, P., Clanet, C. & Limat, L. 2004 Transonic liquid bells. Phys. Fluids 16, 2668–2678.Google Scholar
Coppola, G., De Rosa, F. & de Luca, L. 2013 Surface tension effects on the motion of a free-falling liquid sheet. Phys. Fluids 25 (6), 062103.CrossRefGoogle Scholar
Coppola, G. & de Luca, L. 2006 On transient growth oscillations in linear models. Phys. Fluids 18 (7), 078104.Google Scholar
Crookston, B. & Tullis, B. 2013 Hydraulic design and analysis of labyrinth weirs. II: nappe aeration, instability, and vibration. J. Irrig. Drain Engng 139 (5), 371–377.Google Scholar
de Luca, L. 1999 Experimental investigation of the global instability of plane sheet flows. J. Fluid Mech. 399, 355–376.CrossRefGoogle Scholar
de Luca, L. & Costa, M. 1997a Instability of a spatially developing liquid sheet. J. Fluid Mech. 331, 127–144.CrossRefGoogle Scholar
de Luca, L. & Costa, M. 1997b Stationary waves on plane liquid sheets falling vertically. Eur. J. Mech. (B/Fluids) 16 (1), 75–88.Google Scholar
de Luca, L. & Meola, C. 1995 Surfactant effects on the dynamics of a thin liquid sheet. J. Fluid Mech. 300, 71–85.CrossRefGoogle Scholar
De Rosa, F.2013 Dinamiche instazionarie ed instabilita’ lineare globale di getti liquidi. PhD thesis, Universita’ di Napoli Federico II, http://www.fedoa.unina.it/9459/.Google Scholar
De Rosa, F., Girfoglio, M. & de Luca, L. 2014 Global dynamics analysis of nappe oscillation. Phys. Fluids 26 (12), 122109.CrossRefGoogle Scholar
Finnicum, D. S., Weinstein, S. J. & Rushak, K. J. 1993 The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity. J. Fluid Mech. 255, 647–665.Google Scholar
Howison, S. 2005 Practical Applied Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Joseph, D. D., Funada, T. & Wang, J. 2007 Potential Flows of Viscous and Viscoelastic Fluids. Cambridge University Press.CrossRefGoogle Scholar
Kinsler, L. E., Frey, A. R., Coppens, A. B. & Sanders, J. V. 2000 Fundamentals of Acoustics. John Wiley & Sons.Google Scholar
Le Grand, N., Brunet, P., Lebon, L. & Limat, L. 2006 Propagating waves pattern in a falling liquid curtain. Phys. Rev. E 74, 026305.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lin, S. P., Lian, Z. W. & Creighton, B. J. 1990 Absolute and convective instability of a liquid sheet. J. Fluid Mech. 220, 673–689.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1997 Viscous dissipation in steep capillary-gravity waves. J. Fluid Mech. 344, 271–289.CrossRefGoogle Scholar
Mehring, C. & Sirignano, W. A. 1999 Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets. J. Fluid Mech. 388 (69), 69–113.Google Scholar
Mori, H., Nagamine, T., Ito, R. & Sato, Y. 2012 Mechanism of self-excited vibration of a falling water sheet. Trans. Japan. Soc. Mech. Engrs C 78 (792), 2720–2732.Google Scholar
Naudascher, E. & Rockwell, D. 1994 Flow-Induced Vibrations: An Engineering Guide. Dover.Google Scholar
Ramos, J. I. 2001 Singularities and stability of inviscid, planar liquid membranes. Intl J. Engng Sci. 39, 1935–1948.Google Scholar
Ramos, J. I. 2003 Oscillatory dynamics of inviscid planar liquid sheets. Appl. Maths Comput. 143, 109–144.CrossRefGoogle Scholar
Sato, Y., Miura, S., Nagamine, T., Morii, S. & Ohkubo, S. 2007 Behavior of a falling water sheet. J. Environ. Engng 2, 394–406.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2002 On the stability of a falling liquid curtain. J. Fluid Mech. 463, 163–171.CrossRefGoogle Scholar
Schwartz, H. I. 1964 Nappe oscillation. J. Hydraul. Div. 90 (6), 129–143.CrossRefGoogle Scholar
Weinstein, S. J., Clarke, A., Moon, A. G. & Simister, E. A. 1997 Time-dependent equations governing the shape of a two-dimensional liquid curtain. Part 1. Theory. Phys. Fluids 9, 3625–3636.Google Scholar
Supplementary material: File

Download undefined(File)
File 8.9 MB

Girfoglio et al. supplementary movie

Transition from an initial impulse to the rise of capillary waves

Download Girfoglio et al. supplementary movie(Video)
Video 1.5 MB