Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T02:38:54.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolute and convective instabilities in counter-current gas–liquid film flows

Published online by Cambridge University Press:  11 December 2014

Rajagopal Vellingiri ,
Dmitri Tseluiko  and
Serafim Kalliadasis
Rajagopal Vellingiri
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Dmitri Tseluiko
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
Serafim Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a thin liquid film flowing down an inclined plate in the presence of a counter-current turbulent gas. By making appropriate assumptions, Tseluiko & Kalliadasis (J. Fluid Mech., vol. 673, 2011, pp. 19–59) developed low-dimensional non-local models for the liquid problem, namely a long-wave (LW) model and a weighted integral-boundary-layer (WIBL) model, which incorporate the effect of the turbulent gas. By utilising these models, along with the Orr–Sommerfeld problem formulated using the full governing equations for the liquid phase and associated boundary conditions, we explore the linear stability of the gas–liquid system. In addition, we devise a generalised methodology to investigate absolute and convective instabilities in the non-local equations describing the gas–liquid flow. We observe that at low gas flow rates, the system is convectively unstable with the localised disturbances being convected downwards. As the gas flow rate is increased, the instability becomes absolute and localised disturbances spread across the whole domain. As the gas flow rate is further increased, the system again becomes convectively unstable with the localised disturbances propagating upwards. We find that the upper limit of the absolute instability region is close to the ‘flooding’ point associated with the appearance of large-amplitude standing waves, as obtained in Tseluiko & Kalliadasis (J. Fluid Mech., vol. 673, 2011, pp. 19–59), and our analysis can therefore be used to predict the onset of flooding. We also find that an increase in the angle of inclination of the channel requires an increased gas flow rate for the onset of absolute instability. We generally find good agreement between the results obtained using the full equations and the reduced models. Moreover, we find that the WIBL model generally provides better agreement with the results for the full equations than the LW model. Such an analysis is important for an understanding of the ranges of validity of the reduced model equations. In addition, a comparison of our theoretical predictions with the experiments of Zapke & Kröger (Intl J. Multiphase Flow, vol. 26, 2000, pp. 1439–1455) shows a fairly good agreement. We supplement our stability analysis with time-dependent computations of the linearised WIBL model. To provide some insight into the mechanisms of instability, we perform an energy budget analysis.

JFM classification

Instability: Absolute/convective instability Multiphase and Particle-laden Flows: Gas/liquid flow Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 763 , 25 January 2015 , pp. 166 - 201
Check for updates
Copyright
© 2014 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

Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1994 Wave Flow of Liquid Films. Begell House.Google Scholar
Ali, A., Vafai, K. & Khaled, A.-R. A. 2003 Comparative study between parallel and counter flow configurations between air and falling film desiccant in the presence of nanoparticle suspensions. Intl J. Energy Res. 27, 725–745.Google Scholar
Azzopardi, B. J. 2006 Gas–Liquid Flows. Begell House.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Boomkamp, P. A. M., Boersma, B. J., Miesen, R. H. M. & Beijnon, G. V. 1997 A Chebyshev collocation method for solving two-phase flow stability problems. J. Comput. Phys. 132, 191–200.Google Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.Google Scholar
Brevdo, L., Laure, P., Dias, F. & Bridges, T. J. 1999 Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71.Google Scholar
Briggs, R. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Camassa, R., Forest, M. G., Lee, L., Ogrosky, H. R. & Olander, J. 2012 Ring waves as a mass transport mechanism in air-driven core–annular flows. Phys. Rev. E 86, 066305.CrossRefGoogle ScholarPubMed
Cetinbudaklar, A. G. & Jameson, G. J. 1969 The mechanism of flooding in vertical countercurrent two-phase flow. Chem. Engng Sci. 24, 1669–1680.Google Scholar
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kalaidin, E. N. 1995 Interaction dynamics of solitary waves on a falling film. J. Fluid Mech. 294, 123–154.Google Scholar
Chaté, H. & Manneville, P. 1996 Phase diagram of the two-dimensional complex Ginzburg–Landau equation. Physica A 224, 348–368.Google Scholar
Clift, R., Pritchard, C. L. & Nedderman, R. M. 1966 The effect of viscosity on the flooding conditions in wetted wall columns. Chem. Engng Sci. 21, 87–95.Google Scholar
Craik, A. D. D. 1966 Wind generated waves in thin liquid films. J. Fluid Mech. 26, 369–392.Google Scholar
Demekhin, E. A. 1981 Nonlinear waves in a liquid film entrained by a turbulent gas stream. Fluid Dyn. 16, 188–193.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2013 Wavy liquid films in interaction with a confined laminar gas flow. J. Fluid Mech. 722, 348–393.Google Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2007 Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett. 98, 244502.Google Scholar
Fokas, A. S. & Papageorgiou, D. T. 2005 Absolute and convective instability for evolution PDEs on the half-line. Stud. Appl. Maths 114, 95–114.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Hanratty, T. J. & Engen, J. M. 1957 Interaction between a turbulent air stream and a moving water surface. AIChE J. 3, 299–304.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528.Google Scholar
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.). Cambridge University Press.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Joo, S. W. & Davis, S. H. 1992 Instabilities of three-dimensional viscous falling films. J. Fluid Mech. 242, 529–547.Google Scholar
Juniper, M. P. 2006 The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171–195.Google Scholar
Kabov, O. A., Lyulin, Yu. V., Marchuk, I. V. & Zaitsev, D. V. 2007 Locally heated shear-driven liquid films in microchannels and minichannels. Intl J. Heat Fluid Flow 28, 103–112.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, Springer Series on Applied Mathematical Sciences, vol. 176. Springer.Google Scholar
Kapitza, P. L. 1948 I. Free flow: II. Fluid flow in the presence of continuous gas flow and heat transfer. In Collected Papers of P. L. Kapitza (1965) (ed. Ter Haar, D.), pp. 662–689. Pergamon.Google Scholar
Kelly, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.Google Scholar
Lin, S. P. 2003 Breakup of Liquid Sheets and Jets. Cambridge University Press.Google Scholar
Lin, T.-S. & Kondic, L. 2010 Thin films flowing down inverted substrates: two dimensional flow. Phys. Fluids 22, 052105.Google Scholar
Lin, T.-S., Tseluiko, D. & Kalliadasis, S. 2014 Numerical study of a non-local weakly nonlinear model for a liquid film sheared by a turbulent gas. Procedia IUTAM 11, 98–109.Google Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.Google Scholar
Malamataris, N. A. & Balakotaiah, V. 2008 Flow structure underneath the large amplitude waves of a vertically falling film. AIChE J. 54, 1725–1740.Google Scholar
Miesen, R. & Boersma, B. J. 1995 Hydrodynamic stability of a sheared liquid film. J. Fluid Mech. 301, 175–202.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.CrossRefGoogle Scholar
Mouza, A. A., Pantzali, M. N. & Paras, S. V. 2005 Falling film and flooding phenomena in small diameter vertical tubes: the influence of liquid properties. Chem. Engng Sci. 60, 4981–4991.Google Scholar
Ó Náraigh, L., Spelt, P. D. M. & Shaw, S. J. 2013 Absolute linear instability in laminar and turbulent gas–liquid two-layer channel flow. J. Fluid Mech. 714, 58–94.Google Scholar
Ó Náraigh, L., Spelt, P. D. M. & Zaki, T. A. 2011 Turbulent flow over a liquid layer revisited: multi-equation turbulence modelling. J. Fluid Mech. 683, 357–394.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Pereira, A. & Kalliadasis, S. 2008 Dynamics of a falling film with solutal Marangoni effect. Phys. Rev. E 78, 036312.Google Scholar
Ruyer-Quil, C. & Kalliadasis, S. 2012 Wavy regimes of film flow down a fiber. Phys. Rev. E 85, 046302.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Ruyer-Quil, C., Scheid, B., Kalliadasis, S., Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation. J. Fluid Mech. 538, 199–222.Google Scholar
Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. 2008 Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.Google Scholar
Schlichting, H. 2000 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Semyonov, P. A. 1944 Flows of thin liquid films. J. Tech. Phys. 14, 427–437 (in Russian).Google Scholar
Shearer, C. J. & Davidson, J. F. 1965 The investigation of a standing wave due to gas blowing upwards over a liquid film; its relation to flooding in wetted-wall columns. J. Fluid Mech. 22, 321–335.Google Scholar
Smith, M. K. & Davis, S. H. 1982 The instability of sheared liquid layers. J. Fluid Mech. 121, 187–206.Google Scholar
Suslov, S. A. 2006 Numerical aspects of searching convective/absolute instability transition. J. Comput. Phys. 212, 188–217.Google Scholar
Thorsness, C. B., Morrisroe, P. E. & Hanratty, T. J. 1978 A comparison of linear theory with measurements of the variation of shear stress along a solid wave. Chem. Engng Sci. 33, 579–592.CrossRefGoogle Scholar
Tobias, S. M., Proctor, M. R. E. & Knobloch, E. 1998 Convective and absolute instabilities of fluid flows in finite geometry. Physica D 113, 43–72.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.Google Scholar
Trevelyan, P. M. J. & Kalliadasis, S. 2004a Dynamics of a reactive falling film at large Péclet numbers. II. Nonlinear waves far from criticality: Integral-boundary-layer approximation. Phys. Fluids 16, 3209–3226.Google Scholar
Trevelyan, P. M. J. & Kalliadasis, S. 2004b Wave dynamics of a thin-liquid film falling down a heated wall. J. Engng Maths 50, 177–208.Google Scholar
Trevelyan, P. M. J., Scheid, B., Ruyer-Quil, C. & Kalliadasis, S. 2007 Heated falling films. J. Fluid Mech. 592, 295–334.Google Scholar
Trifonov, Yu. Ya. 2010 Counter-current gas–liquid wavy film flow between the vertical plates analyzed using the Navier–Stokes equations. AIChE J. 56, 1975–1987.Google Scholar
Tseluiko, D. & Kalliadasis, S. 2011 Nonlinear waves in counter-current gas–liquid film flow. J. Fluid Mech. 673, 19–59.Google Scholar
Tseluiko, D., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F. & Kalliadasis, S. 2010 Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory. Physica D 239, 2000–2010.Google Scholar
Vellingiri, R., Tseluiko, D., Savva, N. & Kalliadasis, S. 2013 Dynamics of a liquid film sheared by a co-flowing turbulent gas. Intl J. Multiphase Flow 56, 93–104.Google Scholar
Vlachogiannis, M. & Bontozoglou, V. 2001 Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215.Google Scholar
Woodmansee, D. E. & Hanratty, T. J. 1969 Mechanism for the removal of droplets from a liquid surface by a parallel air flow. Chem. Engng Sci. 24, 299–307.Google Scholar
Yatsyshin, P., Savva, N. & Kalliadasis, S. 2012 Spectral methods for the equations of classical density-functional theory: Relaxation dynamics of microscopic films. J. Chem. Phys. 136, 124113.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Zapke, A. & Kröger, D. G. 2000a Countercurrent gas–liquid flow in inclined and vertical ducts – I: flow patterns, pressure drop characteristics and flooding. Intl J. Multiphase Flow 26, 1439–1455.Google Scholar
Zapke, A. & Kröger, D. G. 2000b Countercurrent gas–liquid flow in inclined and vertical ducts – II: the validity of the Froude–Ohnesorge number correlation for flooding. Intl J. Multiphase Flow 26, 1457–1468.Google Scholar