Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T09:24:58.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial mode interactions in horizontal gas—liquid flows

Published online by Cambridge University Press:  26 April 2006

L. A. Jurman ,
S. E. Deutsch  and
M. J. Mccready
L. A. Jurman
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA Current address: Rohm and Haas Co., PO Box 584, Bristol, PA 19007, USA.
S. E. Deutsch
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA Current address: Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA.
M. J. Mccready
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The behaviour of shear-generated interfacial waves in a cocurrent gas-liquid flow in a small rectangular channel is studied experimentally at conditions close to neutral stability. It is found that the linearly most unstable mode, which typically has a frequency of 8–10 Hz and a wavelength 1–4 cm, grows initially — followed immediately by the first overtone. Measurements of the bicoherence spectrum indicate that the overtone and fundamental are coherent in phase, which suggests that energy is transferred from the fundamental to the linearly stable first overtone. This energy transfer mechanism can stabilize the system, as evidenced by data, which shows that the first mode can saturate with a wave slope small as as 0.005. Theory based on weakly nonlinear mode-interaction equations suggests that this steady state should be stable at conditions close to neutral stability where only overtone modes are present. However, under more severe conditions, where the amplitude of the fundamental mode becomes sufficiently large, a subharmonic mode may be excited. The generation of the subharmonic, when it is linearly stable with respect to the flat film base state, can be interpreted as a linear instability of the steady state containing the fundamental and overtones. Modes that are sidebands (with wavenumbers = k ± δk) to the main peak may also occur. These can participate in interactions with low-frequency modes (i.e. δk) and thereby transfer energy to frequencies much below the fundamental. It is expected that all of these interactions play important roles in determining the wave spectrum of conditions far away from neutral stability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 187 - 219
Copyright
© 1992 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.)

References

Abrams, J. 1984 Turbulent flow over small amplitude solid waves. Ph.D. thesis, Department of Chemical Engineering, University of Illinois, Urbana.
Bannerjee, P. P. & Korpel, A. 1982 Subharmonic generation by resonant three-wave interaction of deep-water capillary waves. Phys. Fluids 25, 1938–1943.Google Scholar
Bendat, J. S. & Piersol, A. G. 1986 Random Data: Analysis and Measurement Procedure, 2nd edn. John Wiley.
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Blennerhassett, P. J. 1980 On the generation of waves by wind. Proc. R. Soc. Lond. A 298, 451–494.Google Scholar
Blennerhassett, P. J. & Smith, F. T. 1987 Short-scale waves on wind-driven water (‘cat's paws’). Proc. R. Soc. Lond. A 410, 1–17.Google Scholar
Bontozoglou, V. & Hanratty, T. J. 1990 Capillary—gravity Kelvin—Helmholtz waves close to resonance. J. Fluid Mech. 217, 71–91.Google Scholar
Bruno, K. 1988 The study of interfacial waves in gas-liquid flows. Ph.D. thesis, University of Notre Dame.
Bruno, K. & McCready, M. J. 1988 Origin of roll waves in gas—liquid flows. AIChE J. 34, 1431–1440.Google Scholar
Bruno, K. & McCready, M. J. 1989 Processes which control the interfacial wave spectrum in separated gas-liquid flows. Intl J. Multiphase Flow 15, 531–552.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity—capillary waves on deep water — I. Weakly nonlinear waves. Stud. Appl. Maths 62, 95–111.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 spatially periodic states. Phys. Fluids (to appear).Google Scholar
Choi, I. 1977 Contributions a l'etude des mechanisms physiques de la generation des ondes de capillarite-gravite a une interface air-eau. Thesis, Université D'Aix Marseille.
Cohen, L. S. & Hanratty, T. J. 1965 Generation of waves in the concurrent flow of air and a liquid. AIChE J. 11, 138–144.Google Scholar
Craik, A. D. D. 1966 Wind-generated waves in thin liquid films. J. Fluid Mech. 26, 369–392.Google Scholar
Ekhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.
Elgar, S. & Guza, R. T. 1985 Observations of bispectra of shoaling surface gravity waves. J. Fluid Mech. 161, 425–448.Google Scholar
Gastel, K. van, Janssen, P. A. E. M. & Komen, G. J. 1985 On phase velocity and growth rate of wind-induced gravity-capillary waves. J. Fluid Mech. 161, 199–216.Google Scholar
Gastel, K. van 1987 Nonlinear interactions of gravity-capillary waves: Lagrangian theory on effects on the spectrum. J. Fluid Mech. 182, 499–523.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hanratty, T. J. 1983 Interfacial instabilities caused by air flow over a thin liquid layer. In: Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 221–259. Academic.
Henderson, D. M. & Hammack, J. L. 1987 Experiments on ripple instabilities. Part 1. Resonant triads J. Fluid Mech. 184, 15–41.Google Scholar
Huerre, P. 1987 Spatio-temporal instabilities in closed and open flows. In Instabilities and Nonequilibrium structures (ed. E. Tirapegui & D. Villarroel), pp. 141–177. Reidel.
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Hughes, D. W. & Proctor, M. R. E. 1990 Chaos and the effect of noise in a model of three-wave mode coupling. Physica D 46, 163–176.Google Scholar
Janssen, P. A. E. M. 1986 The period-doubling of gravity—capillary waves. J. Fluid Mech. 172, 531–546.Google Scholar
Janssen, P. A. E. M. 1987 The initial evolution of gravity—capillary waves. J. Fluid Mech. 184, 581–597.Google Scholar
Jurman, L. A. 1990 Interfacial waves on sheared, thin liquid films. Ph.D. thesis, University of Notre Dame.
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.Google Scholar
Kawai, S. 1979 Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech. 93, 661–703.Google Scholar
Kim, Y. Y. & Hanratty, T. J. 1971 Weak quadratic interactions of two-dimensional waves. J. Fluid Mech. 50, 107–132.Google Scholar
Kim, Y. C. & Powers, E. J. 1979 Digital bispectral analysis and its applications to nonlinear wave interactions. IEEE Trans. Plasma Sci. PS-7, 120–131.Google Scholar
Lange, C. G. & Newell, A. C. 1974 A stability criterion for envelope equations. SIAM J. Appl. Maths 27, 441–456.Google Scholar
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429.Google Scholar
Lleonart, G. T. & Blackman, D. R. 1980 The spectral characteristics of wind-generated capillary waves. J. Fluid Mech 97, 455–479.Google Scholar
McCready, M. J. 1986 Spectral behavior of capillary waves in gas—liquid flows. Phys. Fluids 29, 2836–2842.Google Scholar
McGoldrick, L. F. 1965 Resonant interactions among capillary—gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
McGoldrick, L. F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.Google Scholar
Miya, M., Woodmansee, D. E. & Hanratty, T. J. 1971 A model for roll waves in gas—liquid flows. Chem. Engng Sci. 26, 1915–1931.Google Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in windgenerated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
Prokopiou, Th., McCready, M. J. & Chang, H.-C. 1992 Wave transitions on horizontal gas sheared liquid films. J. Fluid Mech. (submitted).Google Scholar
Qian, J. 1983 Variational approach to the closure problem of turbulence theory. Phys. Fluids 26, 2098–2108.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551–575.Google Scholar
Stuart, J. T. & Diprima, R. C. 1978 The Eckhaus and Benjamin—Feir resonance mechanisms. Proc. R. Soc. Lond. A 362, 27–41.Google 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
Valenzuela, G. R. 1976 The growth of gravity—capillary, waves in the coupled shear flow, J. Fluid Mech. 76, 229–250.Google Scholar
Valenzuela, G. R. & Laing, M. B. 1972 Nonlinear energy transfer in gravity—capillary wave spectra, with applications. J. Fluid Mech. 54, 507–520.Google Scholar
Vyshkind, S. Ya. & Rabinovich, M. I. 1976 The phase stochastization mechanism and the structure of wave turbulence in dissipative media. Sov. Phys. JETP 44, 292–299.Google Scholar
Wersinger, J.-M., Finn, J. M. & Ott, E. 1980 Bifurcation and “strange” behaviour in instability saturation by nonlinear three-wave mode coupling. Phys. Fluids 23, 1142–1154.Google Scholar
Wiggins, S. 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer.