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Global linear stability analysis of falling films with inlet and outlet

Published online by Cambridge University Press:  24 March 2014

C. Albert ,
A. Tezuka  and
D. Bothe
C. Albert
Affiliation:
DFG International Research Training Group 1529, Mathematical Fluid Dynamics, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
A. Tezuka
Affiliation:
Department of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo 169-8555, Japan
D. Bothe*
Affiliation:
Technische Universität Darmstadt, Center of Smart Interfaces and Department of Mathematics, Alarich-Weiss-Straße 10, 64287 Darmstadt, Germany
*
†Email address for correspondence: bothe@csi.tu-darmstadt.de
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Abstract

In this paper, the stability of falling films with different flow conditions at the inlet is studied. This is done with an algorithm for the numerical investigation of stability of steady-state solutions to dynamical systems, which is based on an Arnoldi-type iteration. It is shown how this algorithm can be applied to free boundary problems in hydrodynamics. A volume-of-fluid solver is employed to predict the time evolution of perturbations to the steady state. The method is validated by comparison to data from temporal and spatial stability theory, and to experimental results. The algorithm is used to analyse the flow fields of falling films with inlet and outlet, taking the inhomogeneity caused by different inlet conditions into account. In particular, steady states with a curved interface are analysed. A variety of reasonable inlet conditions is investigated. The instability of the film is convective and perturbations at the inlet could be of importance since they are exponentially amplified as they are transported downstream. However, the employed algorithm shows that there is no significant effect of the inlet condition. It is concluded that the flow characteristics of falling films are stable with respect to the considered time-independent inlet conditions.

JFM classification

Instability: Absolute/convective instability Mathematical Foundations: Computational methods Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014 , pp. 444 - 486
Copyright
© 2014 Cambridge University Press 

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References

Albert, C., Raach, H. & Bothe, D. 2012 Influence of surface tension models on the hydrodynamics of wavy laminar falling films in Volume of Fluid-simulations. Int. J. Multiphase Flow 43, 66–71.CrossRefGoogle Scholar
Alekseenko, S. V., Nakoryakov, V. Y. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31 (9), 1446–1460.Google Scholar
Amaouche, M., Mehidi, N. & Amatousse, N. 2007 Linear stability of a two-layer film flow down an inclined channel: a second-order weighted residual approach. Phys. Fluids 19 (8), 1–14.Google Scholar
Anshus, B. E. 1972 On the asymptotic solution to the falling film stability problem. Ind. Eng. Chem. Fundam. 11 (4), 502–508.CrossRefGoogle Scholar
Anshus, B. E. & Goren, S. L. 1966 A method of getting approximate solutions to the Orr–Sommerfeld equation for flow on a vertical wall. AIChE J. 12 (5), 1004–1008.Google Scholar
Arnoldi, W. E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, 17–29.Google Scholar
Barkley, D., Gomes, M., Gabriela, M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2 (6), 554–573.Google Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.Google Scholar
Bock, D. N. 1977 On the Navier–Stokes equations in noncylindrical domains. J. Differ. Eq. 25, 151–162.CrossRefGoogle Scholar
Bothe, D., Köhne, M. & Prüss, J. 2013 On a class of energy preserving boundary conditions for incompressible Newtonian flows. SIAM J. Math. Anal. 45 (6), 3768–3822.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.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
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Chiba, S. 1998 Global stability analysis of incompressible viscous flow (in Japanese). J. Jpn. Soc. Comput. Fluid Dyn. 7, 20–48.Google Scholar
Christodoulou, K. N. & Scriven, L. E. 1988 Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3, 355–406.Google Scholar
Clément, A. 1996 Coupling of two absorbing boundary conditions for 2D time-domain simulations of free surface gravity waves. J. Comput. Phys. 126 (1), 139–151.CrossRefGoogle Scholar
Dietze, G. F., Leefken, A. & Kneer, R. 2008 Investigation of the backflow phenomenon in falling liquid films. J. Fluid Mech. 595, 435–459.CrossRefGoogle Scholar
Engel, K.-J. & Nagel, R. 2000 In One-Parameter Semigroups for Linear Evolution Equations Graduate Texts in Mathematics, vol. 194, Springer.Google Scholar
Engquist, B. & Majda, A. 1977 Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. USA 74 (5), 1765–1766.Google Scholar
Eriksson, L. E. & Rizzi, A. 1985 Computer-aided analysis of the convergence to steady state of discrete approximations to the euler equations. J. Comput. Phys. 57 (1), 90–128.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
Francois, M. M., Cummins, S. J., Dendy, E. D., Kothe, D. B., Sicilian, J. M. & Williams, M. W. 2006 A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213 (1), 141–173.Google Scholar
Gao, D., Morley, N. B. & Dhir, V. 2003 Numerical simulation of wavy falling film flow using VOF method. J. Comput. Phys. 192 (2), 624–642.CrossRefGoogle Scholar
Graef, M.1966 Über die Eigenschaften zwei-und dreidimensionaler Störungen in Rieselfilmen an geneigten Wänden. Selbstverlag Max-Planck-Institut für Strömungsforschung und Aerodynamische Versuchsanstalt, Göttingen 26.Google Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Jin, G. & Braza, M. 1993 A nonreflecting outlet boundary condition for incompressible unsteady Navier–Stokes calculations. J. Comput. Phys. 107 (2), 239–253.CrossRefGoogle Scholar
Jones, L. O. & Whitaker, S. 1966 An experimental study of falling liquid films. AIChE J. 12 (3), 525–529.CrossRefGoogle Scholar
Kahan, W. 1965 Pracniques: further remarks on reducing truncation errors. Commun. ACM 8 (1), 40–41.Google Scholar
Kato, T. 1995 Perturbation Theory for Linear Operators, Reprint of the 1980 Edition. Springer.Google Scholar
Krantz, W. B. & Goren, S. L. 1971 Stability of thin liquid films flowing down a plane. Ind. Eng. Chem. Fundam. 10 (1), 91–101.Google Scholar
Krantz, W. B. & Owens, W. B. 1973 Spatial formulation of the Orr–Sommerfeld equation for thin liquid films flowing down a plane. AIChE J. 19 (6), 1163–1169.Google Scholar
Lanzerstorfer, D. & Kuhlmann, H. C. 2012 Global stability of the two-dimensional flow over a backward-facing step. J. Fluid Mech. 693, 1–27.CrossRefGoogle Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.Google Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.Google Scholar
Malamataris, N. T. & Papanastasiou, T. C. 1991 Unsteady free surface flows on truncated domains. Indust. Eng. Chem. Res. 30 (9), 2211–2219.CrossRefGoogle Scholar
Nishida, T., Teramoto, Y. & Yoshihara, H. 2005 Hopf bifurcation in viscous incompressible flow down an inclined plane. J. Math. Fluid Mech. 7 (1), 29–71.Google Scholar
Nosoko, T., Yoshimura, P. N., Nagata, T. & Oyakawa, K. 1996 Characteristics of two-dimensional waves on a falling liquid film. Chem. Eng. Sci. 51 (5), 725–732.Google Scholar
Nusselt, W. 1916 Die Oberflächenkondensation des Wasserdampfes. VDI Z. 60, 541–546.Google Scholar
Nusselt, W. 1923 Der Wärmeaustausch am Berieselungskühler. VDI Z. 67, 206–210.Google Scholar
Orlanski, I. 1976 A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys. 21 (3), 251–269.Google Scholar
Pierson, F. W. & Whitaker, S. 1977 Some theoretical and experimental observations of the wave structure of falling liquid films. Ind. Eng. Chem. Fundam. 16 (4), 401–408.Google Scholar
Portalski, S. & Clegg, A. J. 1972 An experimental study of wave inception on falling liquid films. Chem. Eng. Sci. 27 (6), 1257–1265.Google Scholar
Rider, W. J. & Kothe, D. B. 1998 Reconstructing Volume Tracking. J. Comput. Phys. 141 (2), 112–152.Google Scholar
Rieber, M.2004 Numerische Modellierung der Dynamik freier Grenzflächen in Zweiphasenströmungen. PhD thesis, University of Stuttgart.Google Scholar
Rieber, M., Graf, F., Hase, M., Roth, N. & Weigand, B.2000 Numerical simulation of moving spherical and strongly deformed droplets. In Proceedings ILASS-Europe, Darmstadt, 1–6.Google Scholar
Schmid, P. J. & Henningson, D. S.2001 Stability and transition in shear flows, Applied Mathematical Sciences, vol. 142, Springer.Google Scholar
Sharma, A. S., Abdessemed, N., Sherwin, S. J. & Theofilis, V. 2011 Transient growth mechanisms of low Reynolds number flow over a low-pressure turbine blade. Theor. Comput. Fluid Dyn. 25 (1–4), 19–30.Google Scholar
Slattery, J. C. 1999 Advanced Transport Phenomena. Cambridge University Press.Google Scholar
Strobel, W. J. & Whitaker, S. 1969 The effect of surfactants on the flow characteristics of falling liquid films. AIChE J. 15 (4), 527–532.Google Scholar
Tezuka, A. & Suzuki, K. 2006 Three dimensional global linear stability analysis of flow around a spheroid. AIAA J. 44 (8), 1697–1708.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–315.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.CrossRefGoogle Scholar
Tryggvason, G., Bunner, B., Ebrat, O. & Tauber, W.1998 Computations of multiphase flows by a finite difference/front tracking method. I. Multi-fluid flows. In Lecture Series—Von Karman Institute for Fluid Dynamics.Google Scholar
Yih, C. S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Quart. Appl. Math 12 (4), 434–435.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–334.Google Scholar