Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T18:17:27.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Secondary and tertiary excitation of three-dimensional patterns on a falling film

Published online by Cambridge University Press:  26 April 2006

H.-C. Chang ,
M. Cheng ,
E. A. Demekhin  and
D. I. Kopelevich
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
M. Cheng
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
E. A. Demekhin
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent address: Department of Applied Mathematics, Krasnodar Polytechnical Institute. Krasnodar, 350072, Russia.
D. I. Kopelevich
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent address: Department of Applied Mathematics, Krasnodar Polytechnical Institute. Krasnodar, 350072, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The primary instability of a falling film selectively amplifies two-dimensional noise down-stream over three-dimensional modes with transverse variation. If the initial three-dimensional noise is weak or if it has short wavelengths such that they are effectively damped by the capillary mechanism of the primary instability, our earlier study (Chang et al. 1993a) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ2 wave with a solitary wave structure. We show here that the primary transition remains in the presence of significant three-dimensional noise but the secondary transition can be replaced by a selective excitation of oblique triad waves which can even include stable primary disturbances. The resulting secondary checkerboard pattern is associated with a subharmonic mode in the streamwise direction. If the initial transverse noise level is low, a secondary transition to a two-dimensional γ2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 270 , 10 July 1994 , pp. 251 - 276
Copyright
© 1994 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.)

Footnotes

Author to whom all correspondence should be addressed.

References

Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1985 Wave formation on’a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Benney, B. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 1150–1155.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. 1993a Nonlinear evolution of waves on a falling film. J. Fluid Mech. 250, 433–480.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993b Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D 63, 299–320.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993c Construction of stationary waves on a falling film. Comput. Fluid Dyn. 11, 313–322.Google Scholar
Chen, C. C., Lahbabi, A., Chang, H.-C. & Kelley, R. E. 1991 Spanwise pairing of finite-amplitude longitudinal vortex rolls in inclined free-convection boundary layer. J. Fluid Mech. 231, 73–111.Google Scholar
Cheng, M. & Chang, H.-C. 1990 A generalized sideband stability theory via center manifold projection. Phys. Fluids A 2, 1364–1379.Google Scholar
Cheng, M. & Chang, H.-C. 1992 Subharmonic instabilities of finite-amplitude monochromatic waves. Phys. Fluids A 4, 505–523.Google Scholar
Demekhin, E. A., Kaplan, M. A. & Shkadov, V. Ya. 1987 Mathematical models of the theory of viscous liquid films. Izv. Akad. Nauk SSSR, Mekh Zhidk. i Guza 6, 73–81.Google Scholar
Joo, S. W. & Davis, S. H. 1992 Instabilities of three-dimensional viscous falling films. J. Fluid Mech. 242, 529–547.Google Scholar
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 On falling film instabilities and wave breaking. Phys. Fluids A 3, 231–232.Google Scholar
Kuramoto, Y. 1984 Chemical Oscillations, Waves and Turbulence. Springer.
Liu, J., Paul, J. D., Banilower, E. & Gollub, J. P. 1992 Film flow instabilities and spatiotemporal dynamics. In Proc. First Experimental Chaos Conf. (ed. S. Vohra, M. Spano, M. Schlesinger, L. M. Pecora & W. Ditto), pp 225–239. World Scientific.
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurement of the primary instabilities of film flows. J. Fluid Mech. 220, 69–101.Google Scholar
Nepomnyaschy, A. A. 1974a Stability of wave regimes in a film flowing down on inclined plane. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 3, 28–34.Google Scholar
Nepomnyaschy, A. A. 1974b Three-dimensional spatially-periodic motions in liquid films flowing down a vertical plane. Hydrodynamics (Russian) Perm 7, 43–52.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96, 159–205.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, 529–547.Google Scholar
Rosenau, P., Oron, A. & Hyman, J. M. 1992 Bounded and unbounded patterns of the Benney equation. Phys. Fluids A 4, 1102–1104.Google Scholar
Salamon, T. R., Armstrong, R. C. & Brown, R. A. 1994 Traveling waves on vertical films: Numerical analysis by the finite element method. Phys. Fluids (in press).Google Scholar
Thomas, F. O. 1990 An experimental investigation into the role of simultaneous amplitude and phase modulation in the transition of a planar jet. Phys. Fluid A 2, 553–570.Google Scholar