Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T16:11:33.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear waves in counter-current gas–liquid film flow

Published online by Cambridge University Press:  18 February 2011

D. TSELUIKO  and
S. KALLIADASIS
D. TSELUIKO
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
S. 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 investigate the dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film. The solution of the full system of equations describing the gas–liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to pressure and tangential stresses at the interface imposed by the turbulent gas in closed form. We then analyse the liquid film flow under the influence of these perturbations and derive a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto–Sivashinsky equation with an additional term due to the presence of the turbulent gas. This additional term is dispersive and destabilising (for the counter-current case; stabilizing in the co-current case). We also combine the long-wave approximation with a weighted-residual technique to obtain an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. We confirm the existence of large-amplitude stationary waves by computing periodic travelling waves for the integral-boundary-layer approximation and we corroborate our travelling-wave results by time-dependent computations.

JFM classification

Multiphase and Particle-laden Flows: Gas/liquid flow Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011 , pp. 19 - 59
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

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
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier Scientific.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993 Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Demekhin, E. A. 1981 Nonlinear waves in a liquid film entrained by a turbulent gas stream. Fluid Dyn. 16, 188–193.CrossRefGoogle Scholar
Dietze, G. F., Al-Sibai, F. & Kneer, R. 2009 Experimental study of flow separation in laminar falling liquid films. J. Fluid Mech. 637, 73–104.Google Scholar
Dietze, G. F., Leefken, A. & Kneer, R. 2008 Investigation of the backflow phenomenon in falling liquid films. J. Fluid Mech. 595, 435–459.Google Scholar
Drosos, E. I. P., Paras, S. V. & Karabelas, A. J. 2006 Counter-current gas–liquid flow in a vertical narrow channel – Liquid film characteristics and flooding phenomena. Intl J. Multiphase Flow 32, 51–81.CrossRefGoogle Scholar
Dukler, A. E. & Smith, L. 1979 Two-phase interactions in countercurrent flow: studies of the flooding mechanism. Annu. Rep. NUREG/CR-0617. U.S. Nuclear Regulatory Commission, Washington, DC.Google Scholar
Duprat, C., Giorgiutti-Dauphiné, F., Tseluiko, D., Saprykin, S. & Kalliadasis, S. 2009 Liquid film coating a fiber as a model system for the formation of bound states in active dispersive–dissipative nonlinear media. Phys. Rev. Lett. 103, 234501.CrossRefGoogle Scholar
Guguchkin, V. V., Demekhin, E. A., Kalugin, G. N., Markovich, É. É. & Pikin, V. G. 1979 Linear and nonlinear stability of combined plane-parallel flow of a film of liquid and gas. Fluid Dyn. 14, 26–31.CrossRefGoogle Scholar
Homsy, G. M. 1974 Model equations for wavy viscous film flow. In Lectures in Applied Mathematics: Nonlinear Wave Motion (ed. Newell, A.), vol. 15, pp. 191–194. AMS Providence.Google Scholar
Jayanti, S., Tokarz, A. & Hewitt, G. F. 1996 Theoretical investigation of the diameter effect on flooding in countercurrent flow. Intl J. Multiphase Flow 22, 307–324.CrossRefGoogle Scholar
Jurman, L. A. & McCready, M. J. 1989 Study of waves on thin liquid films sheared by turbulent gas flows. Phys. Fluids A 1, 522–536.CrossRefGoogle Scholar
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C. & Velarde, M. G. 2003 a Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 492, 303–338.CrossRefGoogle Scholar
Kalliadasis, S., Kiyashko, A. & Demekhin, E. A. 2003 b Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475, 377–408.CrossRefGoogle Scholar
Kalliadasis, S. & Thiele, U. (Ed.) 2007 Thin Films of Soft Matter. Springer.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin layers of a viscous fluid: 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 (Oxford).Google Scholar
Kawahara, T. 1983 Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett. 51, 381–383.CrossRefGoogle Scholar
Larson, T. K., Oh, C. H. & Chapman, J. C. 1994 Flooding in a thin rectangular slit geometry representative of ATR fuel assembly side-plate flow channels. Nucl. Engng Des. 152, 277–285.Google Scholar
Lee, S. C. & Bankoff, S. G. 1984 Parametric effects on the onset of flooding in flat-plate geometries. Intl J. Heat Mass Transfer 27, 1691–1700.Google Scholar
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429.CrossRefGoogle Scholar
Malamataris, N. A. & Balakotaiah, V. 2008 Flow structure underneath the large amplitude waves of a vertically falling film. AIChE J. 54, 1725–1740.CrossRefGoogle Scholar
Maron, D. M. & Dukler, A. E. 1984 Flooding and upward film flow in vertical tubes. II. Speculations on film flow mechanisms. Intl J. Multiphase Flow 10, 599–621.CrossRefGoogle Scholar
McQuillan, K. W., Whalley, P. B. & Hewitt, G. F. 1985 Flooding in veritcal two-phase flow. Intl J. Multiphase Flow 11, 741–760.Google Scholar
Meza, C. E. & Balakotaiah, V. 2008 Modeling and experimental studies of large amplitude waves on vertically falling films. Chem. Engng Sci. 63, 4704–4734.CrossRefGoogle Scholar
Miles, J. W. 1957 On 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
Nepomnyashchy, A. A. 1974 Stability of wave regimes in a film flowing down an inclined plane. Fluid Dyn. 9, 354–359.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636.Google Scholar
Pantzali, M. N., Mouza, A. A. & Paras, S. V. 2008 Counter-current gas–liquid flow and incipient flooding in inclined small diameter tubes. Chem. Engng Sci. 63, 3966–3978.CrossRefGoogle Scholar
Peng, C.-A., Jurman, L. A. & McCready, M. J. 1991 Formation of solitary waves on gas-sheared liquid layers. Intl J. Multiphase Flow 17, 767–782.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 Dynamics of a falling film with solutal Marangoni effect. Phys. Rev. E 78, 036312.CrossRefGoogle ScholarPubMed
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14, 170–183.CrossRefGoogle 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 fiber. J. Fluid Mech. 603, 431–462.Google Scholar
Scheid, B., Ruyer-Quil, C., Kalliadasis, S., Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J. Fluid Mech. 538, 233–244.CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O. A., Legros, J. C. & Colinet, P. 2004 Validity domain of the Benney equation including Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303–335.CrossRefGoogle Scholar
Schlichting, H. 2000 Boundary-Layer Theory. Springer.Google Scholar
Semyonov, P. A. 1944 Flows of thin liquid films. J. Tech. Phys. (in Russian) 14, 427–437.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
Shkadov, V. Ya. 1967 Wave flow regimes of thin layer of viscous fluid subject to gravity. Fluid Dyn. 2, 29–34.CrossRefGoogle Scholar
Stainthorp, F. P. 1967 The effect of co-current and counter-current air flow on the wave properties of falling liquid films. Trans. Inst. Chem. Engrs 45, 372–382.Google Scholar
Sudo, Y. 1996 Mechanisms and effects of predominant parameters regarding limitation of falling water in vertical counter-current two-phase flow. J. Heat Transfer 118, 715–724.CrossRefGoogle 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.Google Scholar
Trevelyan, P. M. J. & Kalliadasis, S. 2004 a 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.CrossRefGoogle Scholar
Trevelyan, P. M. J. & Kalliadasis, S. 2004 b Wave dynamics on a thin-liquid film falling down a heated wall. J. Engng Maths 50, 177–208.CrossRefGoogle Scholar
Trevelyan, P. M. J., Scheid, B., Ruyer-Quil, C. & Kalliadasis, S. 2007 Heated falling films. J. Fluid Mech. 592, 295–334.CrossRefGoogle 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., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F. & Kalliadasis, S. 2010 a Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory. Physica D 239, 2000–2010.Google Scholar
Tseluiko, D., Saprykin, S. & Kalliadasis, S. 2010 b Interaction of solitary pulses in active dispersive–dissipative media. Proc. Est. Acad. Sci. 59, 139–144.CrossRefGoogle Scholar
Yih, C.-H. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Zapke, A. & Kröger, D. G. 2000 a Countercurrent gas–liquid flow in inclined and vertical ducts. I. Flow patterns, pressure drop characteristics and flooding. Intl J. Multiphase Flow 26, 1439–1455.CrossRefGoogle Scholar
Zapke, A. & Kröger, D. G. 2000 b 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
Zilker, D. P., Cook, G. W. & Hanratty, T. J. 1977 Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 1. Non-separated flows. J. Fluid Mech. 82, 29–51.CrossRefGoogle Scholar