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A full-scale numerical study of interfacial instabilities in thin-film flows

Published online by Cambridge University Press:  26 April 2006

B. Ramaswamy ,
S. Chippada  and
S. W. Joo
B. Ramaswamy
Affiliation:
Department of Mechanical and Environmental Engineering, University of California, Santa Barbara CA 93106-5130, USA
S. Chippada
Affiliation:
TICAM, University of Texas, Austin, TX 78712, USA
S. W. Joo
Affiliation:
School of Mechanical Engineering, Yeungnam University, Kyongsan, Korea
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Abstract

Surface wave instabilities in a two-dimensional thin draining film are studied by a direct numerical simulation of the full nonlinear system. A finite element method is used with an arbitrary Lagrangian–Eulerian formulation to handle the moving boundary problem. Both temporal and spatial stability analysis of the finite-amplitude nonlinear wave regimes are done. As the wavenumber is decreased below the linear cut-off wavenumber, supercritical sinusoidal waves occur as reported earlier from weakly nonlinear analysis and experiments. Further reduction in wavenumber makes the Fourier spectrum broad-banded resulting in solitary humps. This transition from nearly sinusoidal permanent waveforms to solitary humps is found to go through a quasi-periodic regime. The phase boundaries for this quasi-periodic regime have been determined through extensive numerical parametric search. Complex wave interaction processes such as wave merging and wave splitting are discussed. In the exhaustive numerical simulations performed in this paper, no wave-breaking tendency was observed, and it is speculated that the complex wave-interaction processes such as wave merging and wave splitting curb the tendency of the film to break.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 325 , 25 October 1996 , pp. 163 - 194
Copyright
© 1996 Cambridge University Press

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References

Alekseenko, S. V., Nakoryakov, V. Ye. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Amsden, A. A. & Hirt, C. W. 1973 YAQUI: an arbitrary Lagrangian—Eulerian computer program for fluid at at all speeds. Los Alamos Scientific Lab. Report, LA-5100.Google Scholar
Anshus, B. E. & Goren, S. L. 1964 A method of getting approximate solutions to the Orr—Sommerfeld equation for flow on a vertical wall. AIChE J. 12, 1004–1008.Google Scholar
Bach, P. & Villadsen, J. 1984 Simulation of the vertical flow of a thin, wavy film using a finite element method. Intl J. Heat Mass Transfer 27, 815–827.Google Scholar
Behr, M. A., Franca, L. P. & Tezduyar, T. E. 1992 Stabilized finite element methods for the velocity—pressure—stress formulation of incompressible flows. University of Minnesota Supercomputer Institute Research Rep. UMSI 92/8.Google Scholar
Bell, J. B., Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier—Stokes equations. J. Comput. Phys. 85, 257–283.Google Scholar
Belytschko, T. B. & Flanagan, D. P. 1982 Finite element methods with user controlled meshes for fluid—structure interaction. Comput. Meth. Appl. Mech. Engng 33, 669–688.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Brauner, N. & Maron, D. M. 1982 Characteristics of inclined thin films. Waviness and the associated mass transfer. Intl J. Heat Mass Transfer 25, 99–110.Google Scholar
Chan, R. K.-C. 1975 A generalized arbitrary Lagrangian Eulerian method for incompressible flows with sharp interfaces. J. Comput. Phys. 17, 311–331.Google Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. Ann. Rev. Fluid Mech. 26, 103–136.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
Cheng, M. & Chang, H.-C. 1995 Competition between subharmonic and sideband secondary instabilities on a falling film. Phys. Fluids 7, 34–54.Google Scholar
Chin, R. W., Abernathy, F. F. & Bertschy, J. R. 1986 Gravity and shear wave stability of free surface flows. Part 1. Numerical calculations. J. Fluid Mech. 168, 501–513.Google Scholar
Chorin, A. J. 1967 A numerical method for solving viscous incompressible flow problems. J. Comput. Phys. 2, 12–26.Google Scholar
Donea, J. 1983 Arbitrary Lagrangian Eulerian finite element methods. In Computational Methods for Transient Analysis (ed. T. B. Belytschko & T. J. R. Hughes), pp. 474–516. John Wiley & Sons.
Donea, J., Giuliani, S. & Laval, H. 1982a Finite element solution of the unsteady Navier—Stokes equations by fractional step method. Comput. Meth. Appl. Mech. Engng 30, 53–73.Google Scholar
Donea, J., Giuliani, S. & Halleux, J. P. 1982b An arbitrary Lagrangian Eulerian finite element method for transient dynamic fluid—structure interactions. Comput. Meth. Appl. Mech. Engng 33, 689–723.Google Scholar
Dressler, R. F. 1949 Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Maths 2, 149–194.Google Scholar
Finlayson, B. A. 1992 Numerical Methods for Problems with Moving Fronts, pp. 434–456. Ravenna Park Publishing Inc., Seattle.
Gjevik, B. 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925.Google Scholar
Glowinski, R. 1986 Splitting methods for the numerical solution of the incompressible Navier—Stokes equations. In Vistas in Applied Mathematics, Optimization and Software (ed. A. V. Balakrishna et al.), pp. 57–95.
Gresho, P. M. 1990 On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly—consistent mass matrix Part 1: Theory. Intl J. Numer. Meth. Fluids 11, 587–620.Google Scholar
Gresho, P. M. 1991 Incompressible fluid dynamics: some fundamental formulation issues. Ann. Rev. Fluid Mech. 23, 413–453.Google Scholar
Gresho, P. M. & Chan, S. T. 1990 On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly—consistent mass matrix Part 2: Implementation. Intl J. Numer. Meth. Fluids 11, 621–660.Google Scholar
Gresho, P. M., Chan, S. T., Christon, M. A. & Hindmarsh, A. C. 1994 A little more on stabilized Q1Q1 for transient viscous incompressible flow. In Proc. Intl Conf. on Monsoon Variablity and Prediction, Trieste, Italy.
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian—Eulerian computing method for all flow speeds. J. Comput. Phys, 14, 227–253.Google Scholar
Ho, L.-W. & Patera, A. T. 1990 A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows. Comput. Meth. Appl. Mech. Engng 80, 355–366.Google Scholar
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 37–45.Google Scholar
Huerta, A. & Liu, W. K. 1988 Viscous flow with large free surface motion. Comput. Meth. Appl. Mech. Engng 69, 277–324.Google Scholar
Hughes, T. J. R., Liu, W. K. & Zimmerman, T. 1981 Lagrangian—Eulerian finite element formulation for incompressible viscous flow. Comput. Meth. Appl. Mech. Engng 29, 329–349.Google Scholar
Joo, S. W. 1995 Surface Wave Instabilities in Thin Films. World Scientific Publishing Co (in press).
Joo, S. W. & Davis, S. H. 1991 On falling—film instabilities and wave breaking. Phys. Fluids A 3, 231–232.Google Scholar
Joo, S. W. & Davis, S. H. 1992 Irregular waves on viscous falling films. Chem. Engng Commun. 118, 111–123.Google Scholar
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230, 117–146.Google Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid:III. Experimental study of undulatory flow conditions. Zh. Exp. Teor. Fiz. 19, 105. Also in Collected Papers of P. L. Kapitza (ed. D. Ter Haar), vol. 2, pp. 690–709, Pergamon (1965).Google Scholar
Kawahara, M. & Ohmiya, K. 1985 Finite element analysis of density flow using the velocity correction method. Intl J. Numer. Meth. Fluids 5, 981–993.Google Scholar
Kheshgi, H. S. & Scriven, L. E. 1987 Disturbed film flow on a vertical plate. Phys. Fluids 30, 990–997.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional—step method to incompressible Navier—Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Knani, S. & Bankoff, G. 1993 Experimental confirmation of Yih—Benjamin linear stability theory for a falling film on a vertical pipe. J. Israel. Inst. Chem. Engng 22, 58–61.Google Scholar
Krantz, W. B. & Goren, S. L. 1971 Stability of thin liquid films flowing down a plane. Indust. Engng Chem. Fundam. 10, 91–101.Google Scholar
Krsihnamoorthy, S. 1996 Instabilities in heated falling films: a full-scale direct numerical simulation. Ph.D. thesis, Mechanical Engineering, Rice University, Houston, TX.
Lacy, C. E., Sheintuck, M. & Dukler, A. E. 1991 Methods of deterministic chaos applied to the flow of thin wavy films. AIChE J. 37, 481–489.Google Scholar
Le, H. & Moin, P. 1991 An improvement of fractional step method for the incompressible Navier—Stokes equations. J. Comput. Phys. 92, 369–379.Google Scholar
Lee, J.-J. & Mei, C. C. 1996 Stationary waves on an inclined sheet of viscous fluid at high Reynolds and moderate Weber numbers. J. Fluid Mech. 307, 191–229.Google Scholar
Lin, S. P. 1969 Finite amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126.Google Scholar
Lin, S. P. & Wang, C. Y. 1985 Modeling wavy film flows. In Encyclopedia of Fluid Mechanics (ed. N. P. Chermemisinoff), vol. 1, pp. 931–951. Gulf.
Liu, J. & Gollub, J. P. 1993 Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70, 2289–2292.Google 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
Liu, J., Schneider, J. B. & Gollub, J. P. 1995 Three dimensional instabilities of film flows. Phys. Fluids 7, 55–67.Google Scholar
Liu, W. K., Chang, H., Chen, J. & Belytschko, T. 1988 Arbitrary Lagrangian Eulerain Petrov—Galerkin finite elements for for nonlinear continua. Comput. Meth. Appl. Mech. Engng 68, 259–310.Google Scholar
Malamataris, N. T. & Papanastasiou, T. C. 1991 Unsteady free surface flows on truncated domains. Indust. Engng Chem. Res. 30, 2211–2219.Google Scholar
Mizukami, A. & Tsuchiya, M. 1984 A finite element method for the three-dimensional non-steady Navier—Stokes equations. Intl J. Numer. Meth. Fluids 4, 349–357.Google Scholar
Nakaya, C. 1975 Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids 18, 1407–1412.Google Scholar
Nusselt, W. Die oberflächenkondensation des wasserdampfes. Z. Ver. Deut. Ing. 60, 541–569.
Oden, J. T. 1992 Theory and implementation of high-order adaptive hp methods for analysis of incompressible viscous flows. In Computational Nonlinear Mechanics in Aerospace Engineering (ed. S. N. Atluri), pp. 321–363. AIAA.
Orlanski, I. 1976 A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys. 21, 251–269.Google Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.
Pierson, F. W. & Whitaker, S. 1977 Some theoretical and experimental observations of the wave structure of falling liquid films. Indust. Engng Chem. Fundam. 16, 401–408.Google Scholar
Pironneau, O. 1982 On the transport—diffusion algorithm and its applications to the Navier—Stokes equations. Numerische Mathematic 38, 309–332.Google Scholar
Portalski, S. & Clegg, A. J. 1972 An experimental study of wave inception on falling liquid films. Chem. Engng Sci. 27, 1257–1265.Google Scholar
Pracht, W. E. 1975 Calculating three-dimensional fluid flows at all speeds with an Eulerian—Lagrangian computing mesh. J. Comput. Phys. 17, 132–159.Google Scholar
Prokopiou, Th., Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665–691.Google Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Ramaswamy, B., Jue, T. C. & Akin, J. E. 1992 Semi-implicit and explicit finite element schemes for coupled fluid/thermal problems. Intl J. Numer. Meth. Engng 34, 675–696.Google Scholar
Ramaswamy, B. & Kawahara, M. 1987a An equal order velocity pressure finite element formulation for solving the time-dependent incompressible Navier—Stokes equations. Bull. Facul. Sci. Engng Chuo Univ. 30, 63–104.Google Scholar
Ramaswamy, B. & Kawahara, M. 1987b Arbitrary Lagrangian—Eulerian finite element method for unsteady, convective, incompressible viscous free surface fluid flow. Intl J. Numer. Meth. Fluids 7, 1053–1075.Google Scholar
Rice, J. G. & Schnipke, R. J. 1986 An equal—order velocity—pressure formulation that does not exhibit spurious pressure modes. Comput. Meth. Appl. Mech. Engng 58, 135–149.Google Scholar
Rosenau, P. & Oron, A. 1989 Evolution and breaking of liquid film flowing on a vertical cylinder. Phys. Fluids A 1, 1763–1766.Google Scholar
Roskes, G. J. 1970 Three-dimensional long waves on a liquid film. Phys. Fluids 13, 1440–1445.Google Scholar
Saito, H. & Scriven, L. E. 1981 Study of the coating flow by the finite element method. J. Comput. Phys. 42, 53–76.Google Scholar
Salamon, T. R., Armstrong, R. C. & Brown, R. A. 1994 Travelling waves on vertical films: Numerical analysis using the finite element method. Phys. Fluids 6, 2202–2220.Google Scholar
Schneider, G. E., Raithby, G. D. & Yovanovich, M. M. 1978 Finite—element solution procedures for solving the incompressible, Navier—Stokes equations using equal order variable interpolation. Numer. Heat Transfer 1, 443–451.Google Scholar
Shaw, C. T. 1991 Using a segregated finite element scheme to solve the incompressible Navier—Stokes equations. Intl J. Numer. Meth. Fluids 12, 81–92.Google Scholar
Sivashinsky, G. I. & Michelson, D. M. 1980 On the irregular wavy flow of liquid film down a vertical plane. Prog. Theor. Phys. 63, 2112.Google Scholar
Soulaïmani, A., Fortin, M., Dhatt, G. & Ouellet, Y. 1991 Finite element simulation of two- and three-dimensional free surface flows. Comput. Meth. Appl. Mech. Engng 86, 265–296.Google Scholar
Temam, R. 1971 On the Theory and Numerical Analysis of the Navier—Stokes Equations. North-Holland.
Whitaker, S. 1964 Effect of surface active agents on stability of falling liquid films. Indust. Engng Chem. Fundam. 3, 132–142.Google Scholar
Van Kan, J. 1986 A second-order accurate pressure correction scheme for viscous incompressible flow. SIAM J. Sci. Statist. Comput. 7, 870–891.Google Scholar
Yih, C.-S. 1955 Stability of parallel laminar flow with a free surface. Proc. 2nd US Congr. Appl. Mech., pp. 623–628. ASME.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–324.Google Scholar
Zienkiewicz, O. C. & Wu, J. 1991 Incompressibility without tears — how to avoid restriction of mixed formulation. Intl J. Numer. Meth. Engng 34, 1189–1203.Google Scholar
Zienkiewicz, O. C. & Wu, J. 1992 A general explicit or semi-explicit algorithm for compressible and incompressible flows. Intl J. Numer. Meth. Engng 35, 457–479.Google Scholar