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Gas-sheared falling liquid films beyond the absolute instability limit

Published online by Cambridge University Press:  25 September 2023

Misa Ishimura Open the ORCID record for Misa Ishimura [Opens in a new window] ,
Sophie Mergui Open the ORCID record for Sophie Mergui [Opens in a new window] ,
Christian Ruyer-Quil Open the ORCID record for Christian Ruyer-Quil [Opens in a new window]  and
Georg F. Dietze Open the ORCID record for Georg F. Dietze [Opens in a new window]
Misa Ishimura
Affiliation:
Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambèry, France Department of Mechanical Engineering, Yokohama National University, Kanagawa 240-8501, Japan
Sophie Mergui
Affiliation:
Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France Sorbonne Université, UFR 919, 4 place Jussieu, F-75252 Paris CEDEX 05, France
Christian Ruyer-Quil
Affiliation:
Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambèry, France
Georg F. Dietze*
Affiliation:
Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France
*
†Email address for correspondence: dietze@fast.u-psud.fr
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Abstract

We study the effect of a confined turbulent counter-current gas flow on the waviness of a weakly inclined falling liquid film. Our study is centred on experiments in a channel of 13 mm height, using water and air, where we have successively increased the counter-current gas flow rate until flooding. Computations with a new low-dimensional model and linear stability calculations are used to elucidate the linear and nonlinear wave dynamics. We find that the gas pressure gradient plays an important role in countering the stabilizing effect of the tangential gas shear stress at the liquid–gas interface. At very low inclination angles, the latter effect dominates and can suppress the long-wave Kapitza instability unconditionally. By contrast, for non-negligible inclination, the gas effect is linearly destabilizing, amplifies the height of nonlinear Kapitza waves, and exacerbates coalescence-induced formation of large-amplitude tsunami waves. Kapitza waves do not undergo any catastrophic transformation when the counter-current gas flow rate is increased beyond the absolute instability (AI) limit. On the contrary, we find that AI is an effective linear wave selection mechanism in a noise-driven wave evolution scenario, leading to highly regular downward-travelling nonlinear wave trains, which preclude coalescence events. In our experiments, where Kapitza waves develop in a protected region before coming into contact with the gas, flooding is eventually caused far beyond the AI limit by upward-travelling short-wave ripples. Based on our linear stability calculations for arbitrary wavenumbers, we have uncovered a new short-wave interfacial instability mode with negative linear wave speed, causing these ripples.

JFM classification

Interfacial Flows (free surface): Thin films Multiphase and Particle-laden Flows: Gas/liquid flow Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 971 , 25 September 2023 , A37
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Alekseenko, S.V., Aktershev, S.P., Cherdantsev, A.V., Kharlamov, S.M. & Markovich, D.M. 2009 Primary instabilities of liquid film flow sheared by turbulent gas stream. Intl J. Multiphase Flow 35, 617–627.CrossRefGoogle Scholar
Alekseenko, S.V., Antipin, V.A., Bobylev, A.V. & Markovich, D.M. 2007 Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder. Exp. Fluids 43 (2–3), 197–207.CrossRefGoogle Scholar
Azzopardi, B.J., Mudde, R.F., Lo, S., Morvan, H., Yan, Y. & Zhao, D. 2011 Hydrodynamics of Gas–Liquid Reactors: Normal Operation and Upset Conditions. John Wiley & Sons.CrossRefGoogle Scholar
Bankoff, S.G. & Lee, S.C. 1986 A Critical Review of the Flooding Literature, vol. 2, pp. 95–180. Springer.Google Scholar
Barmak, I., Gelfgat, A., Ullman, A. & Brauner, N. 2016 a Stability of stratified two-phase flows in inclined channels. Phys. Fluids 28, 084101.CrossRefGoogle Scholar
Barmak, I., Gelfgat, A., Vitoshkin, H., Ullman, A. & Brauner, N. 2016 b Stability of stratified two-phase flows in horizontal channels. Phys. Fluids 28, 044101.CrossRefGoogle 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.CrossRefGoogle Scholar
Brooke Benjamin, T. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Brooke Benjamin, T. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.CrossRefGoogle Scholar
Camassa, R., Ogrosky, H.R. & Olander, J. 2017 Viscous film-flow coating the interior of a vertical tube. Part 2. Air-driven flow. J. Fluid Mech. 825, 1056–1090.CrossRefGoogle Scholar
Chang, H.C., Demekhin, E.A. & Kalaidin, E. 1996 a Simulation of noise-driven wave dynamics on a falling film. AIChE J. 42 (6), 1553–1568.CrossRefGoogle Scholar
Chang, H.C., Demekhin, E.A., Kalaidin, E. & Ye, Y. 1996 b Coarsening dynamics of falling-film solitary waves. Phys. Rev. E 54 (2), 1467–1477.CrossRefGoogle ScholarPubMed
Cohen, L.S. & Hanratty, T.J. 1968 Effect of waves at a gas–liquid interface on a turbulent air flow. J. Fluid Mech. 31 (3), 467–479.CrossRefGoogle Scholar
Cohen-Sabban, J., Gaillard-Groleas, J. & Crepin, P.-J. 2001 Quasi-confocal extended field surface sensing. In Proceedings of SPIE (ed. A. Duparre & B. Singh), Optical Metrology Roadmap for the Semiconductor, Optical, and Data Storage Industries II, vol. 4449, pp. 178–183. SPIE.Google 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. 2016 On the Kapitza instability and the generation of capillary waves. J. Fluid Mech. 789, 368–401.CrossRefGoogle Scholar
Dietze, G.F. 2019 Effect of wall corrugations on scalar transfer to a wavy falling liquid film. J. Fluid Mech. 859, 1098–1128.CrossRefGoogle 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.CrossRefGoogle Scholar
Doedel, E.J. 2008 AUTO07P: Continuation and bifurcation software for ordinary differential equations. Montreal Concordia University.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
Fair, J.R. & Bravo, J.L. 1990 Distillation columns containing structured packing. Chem. Engng Prog. 86 (1), 19–29.Google Scholar
Floryan, J.M., Davis, S.H. & Kelly, R.E. 1987 Instabilities of a liquid film flowing down a slightly inclined plane. Phys. Fluids 30 (4), 983–989.CrossRefGoogle Scholar
Frederick, K.A. & Hanratty, T.J. 1988 Velocity measurements for a turbulent nonseparated flow over solid waves. Exp. Fluids 6 (7), 477–486.CrossRefGoogle Scholar
Halpern, D. & Grotberg, J.B. 2003 Nonlinear saturation of the Rayleigh-instability due to oscillatory flow in a liquid-lined tube. J. Fluid Mech. 492, 251–270.CrossRefGoogle Scholar
Hanratty, T.J. & Engen, J.M. 1957 Interaction between a turbulent air stream and a moving water surface. AIChE J. 3 (3), 299–304.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M.G. 2012 Falling Liquid Films, Applied Mathematical Sciences, vol. 176. Springer.CrossRefGoogle Scholar
Kapitza, P.L. 1948 Wave flow of thin layer of viscous fluid (in Russian). Zh. Eksp. Teor. Fiz. 18 (1), 3–28.Google Scholar
Kofman, N. 2014 Films liquides tombants avec ou sans contre-écoulement de gaz: application au problème de l'engorgement dans les colonnes de distillation. PhD thesis, Université Pierre et Marie Curie.Google Scholar
Kofman, N., Mergui, S. & Ruyer-Quil, C. 2014 Three-dimensional instabilities of quasi-solitary waves in a falling liquid film. J. Fluid Mech. 757, 854–887.CrossRefGoogle Scholar
Kofman, N., Mergui, S. & Ruyer-Quil, C. 2017 Characteristics of solitary waves on a falling liquid film sheared by a turbulent counter-current gas flow. Intl J. Multiphase Flow 95, 22–34.CrossRefGoogle Scholar
Kupfer, K., Bers, A. & Ram, A.K. 1987 The cusp map in the complex-frequency plane for absolute instabilities. Phys. Fluids 30 (10), 3075–3082.CrossRefGoogle Scholar
Kushnir, R., Barmak, I., Ullmann, A. & Brauner, N. 2021 Stability of gravity-driven thin-film flow in the presence of an adjacent gas phase. Intl J. Multiphase Flow 135, 103443.CrossRefGoogle Scholar
Lapkin, A. & Anastas, P.T. (Ed.) 2018 Handbook of Green Chemistry, Green Chemical Engineering, vol. 12. Wiley-VHC.Google Scholar
Lavalle, G., Grenier, N., Mergui, S. & Dietze, G.F. 2020 Solitary waves on superconfined falling liquid films. Phys. Rev. Fluids 5 (3), 032001(R).CrossRefGoogle Scholar
Lavalle, G., Li, Y., Mergui, S., Grenier, N. & Dietze, G.F. 2019 Suppression of the Kapitza instability in confined falling liquid films. J. Fluid Mech. 860, 608–639.CrossRefGoogle Scholar
Lavalle, G., Mergui, S., Grenier, N. & Dietze, G.F. 2021 Superconfined falling liquid films: linear versus nonlinear dynamics. J. Fluid Mech. 919, R2.CrossRefGoogle Scholar
Lel, V.V., Al-Sibai, F., Leefken, A. & Renz, U. 2005 Local thickness and wave velocity measurement of wavy films with a chromatic confocal imaging method and a fluorescence intensity technique. Exp. Fluids 39 (5), 856–864.CrossRefGoogle Scholar
Liu, J. & Gollub, J.P. 1993 Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70 (15), 2289–2292.CrossRefGoogle ScholarPubMed
Liu, J. & Gollub, J.P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6 (5), 1702–1712.CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2019 On the large difference between Benjamin's and Hanratty's formulations of perturbed flow over uneven terrain. J. Fluid Mech. 871, 534–561.CrossRefGoogle Scholar
MATLAB 2015 Version 8.6 (R2015b). The MathWorks.Google Scholar
McCready, M.J. & Chang, H.-C. 1994 Formation of large disturbances on sheared and falling liquid films. Chem. Engng Commun. 141–142 (1), 347–358.Google Scholar
Mergui, S., Lavalle, G., Li, Y., Grenier, N. & Dietze, G.F. 2023 Nonlinear dynamics of strongly-confined gas-sheared falling liquid films. J. Fluid Mech. 954, A19.CrossRefGoogle 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
Miesen, R. & Boersma, B.J. 1995 Hydrodynamic stability of a sheared liquid film. J. Fluid Mech. 301, 175–202.CrossRefGoogle Scholar
Miles, J.W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.CrossRefGoogle Scholar
Miyara, A. 1999 Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves. Heat Mass Transfer 35, 298–306.CrossRefGoogle Scholar
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic Schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46, 1021–1036.CrossRefGoogle Scholar
Náraigh, L.Ó., Spelt, P.D.M., Matar, O.K. & Zaki, T.A. 2011 Interfacial instability in turbulent flow over a liquid film in a channel. Intl J. Multiphase Flow 37 (7), 812–830.CrossRefGoogle Scholar
Oron, A. & Gottlieb, O. 2004 Subcritical and supercritical bifurcations of the first- and second-order Benney equations. J. Engng Maths 50 (2–3), 121–140.CrossRefGoogle Scholar
Orszag, S.A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689–703.CrossRefGoogle Scholar
Özgen, S., Carbonaro, M. & Sarma, G.S.R. 2002 Experimental study of wave characteristics on a thin layer of de/anti-icing fluid. Phys. Fluids 14 (10), 3391–3402.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prandtl, L. 1925 Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. Angew. Math. Mech. 5, 136–139.CrossRefGoogle Scholar
Richard, G., Ruyer-Quil, C. & Vila, J.P. 2016 A three-equation model for thin films down an inclined plane. J. Fluid Mech. 804, 162–200.CrossRefGoogle Scholar
Russo, S. & Luchini, P. 2016 The linear response of turbulent flow to a volume force: comparison between eddy-viscosity model and DNS. J. Fluid Mech. 790, 104–127.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6 (2), 277–292.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 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 (1), 170–183.CrossRefGoogle Scholar
Samanta, A. 2014 Shear-imposed falling film. J. Fluid Mech. 753, 131–149.CrossRefGoogle Scholar
Samanta, A. 2020 Optimal disturbance growth in shear-imposed falling film. AIChE J. 66 (5), 0001–1541.CrossRefGoogle Scholar
Schmidt, P., Náraigh, L.Ó., Lucquiaud, M. & Valluri, P. 2016 Linear and nonlinear instability in vertical counter-current laminar gas–liquid flows. Phys. Fluids 28, 042102.CrossRefGoogle Scholar
Shkadov, V.Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn. 2 (1), 29–34.CrossRefGoogle Scholar
Thompson, A.B., Gomes, S.N., Denner, F., Dallaston, M.C. & Kalliadasis, S. 2019 Robust low-dimensional modelling of falling liquid films subject to variable wall heating. J. Fluid Mech. 877, 844–881.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.CrossRefGoogle Scholar
Tilley, B.S., Davis, S.H. & Bankoff, S.G. 1994 Linear stability theory of two-layer fluid flow in an inclined channel. Phys. Fluids 6 (12), 3906–3922.CrossRefGoogle Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Trifonov, Y.Y. 2010 a Counter-current gas–liquid wavy film flow between the vertical plates analyzed using the Navier–Stokes equations. AIChE J. 56 (8), 1975–1987.Google Scholar
Trifonov, Y.Y. 2010 b Flooding in two-phase counter-current flows: numerical investigation of the gas–liquid wavy interface using the Navier–Stokes equations. Intl J. Multiphase Flow 36, 549–557.CrossRefGoogle Scholar
Trifonov, Y.Y. 2017 Instabilities of a gas–liquid flow between two inclined plates analyzed using the Navier–Stokes equations. Intl J. Multiphase Flow 95, 144–154.CrossRefGoogle Scholar
Trifonov, Y.Y. 2019 Nonlinear wavy regimes of a gas–liquid flow between two inclined plates analyzed using the Navier–Stokes equations. Intl J. Multiphase Flow 112, 170–182.CrossRefGoogle Scholar
Tseluiko, D. & Kalliadasis, S. 2011 Nonlinear waves in counter-current gas–liquid film flow. J. Fluid Mech. 673, 19–59.CrossRefGoogle Scholar
Valluri, P., Matar, O.K., Hewitt, G.F. & Mendes, M.A. 2005 Thin film flow over structured packings at moderate Reynolds numbers. Chem. Engng Sci. 60, 1965–1975.CrossRefGoogle Scholar
Van Driest, E.R. 1956 On turbulent flow near a wall. J. Aeronaut. Sci. 23 (11), 1007–1011.CrossRefGoogle Scholar
Vellingiri, R., Tseluiko, D. & Kalliadasis, S. 2015 Absolute and convective instabilities in counter-current gas–liquid film flows. J. Fluid Mech. 763, 166–201.CrossRefGoogle Scholar
Vlachos, N.A., Paras, S.V., Mouza, A.A. & Karabelas, A.J. 2001 Visual observations of flooding in narrow rectangular channels. Intl J. Multiphase Flow 27, 1415–1430.CrossRefGoogle Scholar
Yih, C.S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–334.CrossRefGoogle Scholar
Yoshimura, P.N., Nosoko, P. & Nagata, T. 1996 Enhancement of mass transfer into a falling laminar liquid film by two-dimensional surface waves – some experimental observations and modeling. Chem. Engng Sci. 51 (8), 1231–1240.CrossRefGoogle Scholar
Zapke, A. & Kröger, D.G. 2000 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
Zhou, G. & Prosperetti, A. 2020 Capillary waves on a falling film. Phys. Rev. Fluids 5, 114005.CrossRefGoogle 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 (1), 29–51.CrossRefGoogle Scholar

Ishimura et al. Supplementary Movie 1

Gas-induced wave coalescence on a weakly-inclined falling liquid film subject to a counter-current turbulent gas flow. Parameters according to panel 18b: below the absolute instability limit.

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Video 2.7 MB

Ishimura et al. Supplementary Movie 2

Regular wave train produced by absolute instability (AI) on a weakly-inclined falling liquid film subject to a counter-current turbulent gas flow. Parameters according to panel 18c: beyond the AI limit.

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Video 2.6 MB

Ishimura et al. Supplementary Movie 3

Standing ripples generated by absolute instability on a vertically falling liquid film subject to an increasingly strong counter-current turbulent gas flow. Parameters according to figure 23.

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Video 44.6 MB