Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T21:45:33.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Modulated, frequency-locked, and chaotic cross-waves

Published online by Cambridge University Press:  26 April 2006

William B. Underhill ,
Seth Lichter  and
Andrew J. Bernoff
William B. Underhill
Affiliation:
Department of Aerospace and Mechanical Engineering. The University of Arizona, Tucson, AZ 85721, USA
Seth Lichter
Affiliation:
Department of Aerospace and Mechanical Engineering. The University of Arizona, Tucson, AZ 85721, USA
Andrew J. Bernoff
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, AZ 85721, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Measurements were made of the wave height of periodic, quasi-periodic, and chaotic parametrically forced cross-waves in a long rectangular channel. In general, three frequencies (and their harmonics) may be observed: the subharmonic frequency and two slow temporal modulations — a one-mode instability associated with streamwise variation and a sloshing motion associated with spanwise variation. Their interaction, as forcing frequency, f, and forcing amplitude, a, were varied, produced a pattern of Arnold tongues in which two or three frequencies were locked. The overall picture of frequency-locked and -unlocked regions is explained in terms of the Arnold tongues predicted by the circle-map theory describing weakly coupled oscillators. Some of the observed tongues are apparently folded by a subcritical bifurcation, with the tips of the tongues lying on the unstable manifold folded under the observed stable manifold. Near the intersection of the neutral stability curves for two adjacent modes, a standing wave localized on one side of the tank was observed in agreement with the coupled-mode analysis of Ayanle, Bernoff & Lichter (1990). At large cross-wave amplitudes, the spanwise wave structure apparently breaks up, because of modulational instability, into coherent soliton-like structures that propagate in the spanwise direction and are reflected by the sidewalls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 225 , April 1991 , pp. 371 - 394
Copyright
© 1991 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

Anosov, D. V. & Arnold, V. I., 1988 Dynamical systems I. In Ordinary Differential Equations and Smooth Dynamical Systems. Springer.
Arneodo, A., Coullet, P. H. & Spiegel, E. A., 1983 Cascade of period doublings of tori. Phys. Lett. 94A, 1–6.Google Scholar
Arnson, D. G., Chory, M. A., Hall, G. R. & McGehee, R. P., 1982 Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. Commun. Math. Phys. 83, 303–354.Google Scholar
Ayanle, H., Bernoff, A. J. & Lichter, S., 1990 Spanwise modal competition in cross-waves. Physica D 43, 87–104.Google Scholar
Bak, P., Bohr, T. & Jensen, M. H., 1985 Mode-locking and the transition to chaos in dissipative systems. Phys. Scripta T T9, 50–58.Google Scholar
Barnard, B. J. S. & Pritchard, W. G. 1972 Cross-waves; part 2. Experiments. J. FluidMech. 55, 245–255.Google Scholar
Chen, J. M.: 1987 Subharmonic resonance of nonlinear cross-waves: theory and experiments. PhD thesis, The University of Arizona, Tucson.
Ciliberto, S. & Gollub, J. B., 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922–925.Google Scholar
Ciliberto, S. & Gollub, J. B., 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Coullet, P., Tresser, C. & Arneodo, A., 1980 Transition to turbulence for doubly periodic flows. Phys. Lett. 77A, 327–331.Google Scholar
Cumming, A. & Linsay, P. S., 1988 Quasiperiodicity and chaos in a system with three competing frequencies. Phys. Rev. Lett. 60, 2719–2722.Google Scholar
Ecke, R. E., Farmer, J. D. & Umberger, D. K., 1989 Scaling of the Arnold tongues. Nonlinearity 2, 175–196.Google Scholar
Funakoshi, M. & Inoue, S., 1987 Chaotic behaviour of resonantly forced surface water waves. Phys. Lett. A 121, 229–232.Google Scholar
Funakoshi, M. & Inoue, S., 1988 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.Google Scholar
Grassberger, P. & Procaccia, I., 1983 Measuring the strangeness of strange attractors. Physica 9D, 189–208.Google Scholar
Gu, X. M. & Sethna, P. R., 1987 Resonant surface waves and chaotic phenomena. J. Fluid Mech. 183, 543–565.Google Scholar
Gu, X. M., Sethna, P. R. & Narain, A., 1988 On three-dimensional nonlinear subharmonic resonant surface waves in a fluid: part 1 — theory. J. Appl. Mech. 55, 213–219.Google Scholar
Guckenheimer, J. & Holmes, P., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Chapter 7. Springer.
Jones, A. J.: 1984 The generation of cross-waves in a long deep channel by parametric resonance. J. Fluid Mech. 138, 53–74.Google Scholar
Kit, E. & Shemer, L., 1989 On the neutral stability of cross-waves. Phys. Fluids 1, 1128–1132.Google Scholar
Lichter, S. & Bernoff, A. J., 1988 Stability of steady cross-waves: theory and experiment. Phys. Rev. A 37, 1663–1667.Google Scholar
Lichter, S. & Chen, J., 1987 Subharmonic resonance of nonlinear cross- waves. J. Fluid Mech. 183, 451.Google Scholar
Lichter, S. & Underhill, W. B., 1987 Mode-number shifting of nonlinear cross-waves. Phys. Rev. A 35, 5282–5284.Google Scholar
Linsay, P. S. & Cumming, A., 1989 Three-frequency quasiperiodicity, phase locking and the onset of chaos. Physica D 40, 196–217.Google Scholar
MacKay, R. S. & Tresser, C., 1986 Transition to topological chaos for circle maps. Physica 19D, 206–237.Google Scholar
Miles, J. W.: 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Newhouse, S., Ruelle, D. & Takens, F., 1978 Occurrence of strange axiom A attractors near quasperiodic flows on Tmm [ges ] 3. Commun. Math. Phys. 64, 35–40.Google Scholar
Shemer, L. & Lichter, S., 1987 Identification of cross-wave regimes in the vicinity of a cut-off frequency. Phys. Fluids 30, 3427–3433.Google Scholar
Shemer, L. & Lighter, S., 1990 The mode number dependence of neutral stability of cross-waves. Exps Fluids 9, 148–152.Google Scholar
Simonelli, F. & Gollub, J. P., 1988 Stability boundaries and phase-space measurement for spatially extended dynamical systems. Rev. Sci. Instrum. 59, 280–284.Google Scholar
Takens, F.: 1981 Detecting strange attractors in turbulence. Lecture Notes in Maths, vol. 898, pp. 366–381.Google Scholar
Underhill, W. B.: 1990 Transitions to chaotic cross-waves: An experimental investigation. PhD thesis, The University of Arizona, Tucson.
Van Atta, C. W. & Gharib, M. 1987 Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 174, 113–133.Google Scholar
Virnig, J. C., Berman, A. S. & Sethna, P. R., 1988 On three-dimensional nonlinear subharmonic resonant surface waves in a fluid: part II — experiment. Trans. A8ME 55, 220–224.Google Scholar
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A., 1984 Determining Lyapunov exponents from a time series. Physica 16D, 285–317.Google Scholar
Wu, J., Keolian, R. & Rudnick, I., 1984 Observation of a non-propagating hydrodynamic soliton. Phys. Rev. Lett. 52, 1421–1424.Google Scholar