Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-20T04:16:02.263Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitions to chaos in a forced jet: intermittency, tangent bifurcations and hysteresis

Published online by Cambridge University Press:  26 April 2006

George Broze  and
Fazle Hussain
George Broze
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston. TX 77204–4792, USA
Fazle Hussain
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston. TX 77204–4792, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Experimental studies of a forced transitional jet at moderate Reynolds numbers reveal interesting transitions between low-dimensional states, namely tangent bifurcations, intermittency and hysteresis. The experiments were carried out in an axisymmetric air jet with a laminar top-hat exit profile in the low-noise ambient of a large anechoic chamber, using forcing amplitude and frequency as control parameters. Tangent bifurcations are seen to occur in two different transitions from periodicity to chaos: (i) from stable pairing to nearly periodic modulations of pairing and (ii) from stable double pairing to a quarter-harmonic chaotic attractor. In case (i), an empirically derived mapping closely reproduces the temporal dynamics of one chaotic attractor at a point just after the tangent bifurcation. In case (ii), the intermittency was characterized by estimating the scaling exponent of the p.d.f. of the periodic durations, which was found to be close to the characteristic value for type-II intermittency. Hysteresis is seen at higher Strouhal numbers in the transitions between aperiodic modulations and the periodic/chaotic double pairing states. Based on simultaneous flow visualization and velocimetry, the hysteresis appears to be associated with intermittent tilting of (otherwise) axisymmetric vortices. These transitions are explained in terms of feedback-driven dynamics from vortex roll-ups and pairings, which can be phase-locked (periodic) or unlocked (nearly quasi-periodic and chaotic). The observed transitions connect large regions of deterministic behaviour in the phase diagram, confirming the existence of a low-dimensional dynamical system in transitional jets-an open flow of technological relevance.

Two-point coherence measurements indicate that spatial coupling (and, hence, coherent motion from pairing dynamics) extends for five to eight diameters from the exit, well beyond the locations of pairing and double pairing. This justifies the use of single-point measurements and confirms our hypothesis that the dynamics in this convectively unstable flow are primarily temporal rather than spatio-temporal.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 311 , 25 March 1996 , pp. 37 - 71
Copyright
© 1996 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

Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115.Google Scholar
Bergé, P., Dubois, M., Manneville, P. & Pomeau, Y. 1980 Intermittency in Rayleigh-Bénard Convection. J. Phys. (Paris) Lett. 41, L–344.Google Scholar
Berger, W. D. 1993 Visualization of some vortex interactions in a circular jet. BS senior thesis, University of Houston Mechanical Engineering Dept.
Brandstäter, A. & Swinney, H. L. 1987 A strange attractor in weakly turbulent Couette-Taylor flow.. Phys. Rev. A 35, 2207.Google Scholar
Bridges, J. E. 1990 Application of coherent structure and vortex sound theories to jet noise. PhD dissertation, University of Houston Mechanical Engineering Department.
Bridges, J. E. & Hussain, F. 1992 Direct evaluation of aeroacoustic theory in a jet. J. Fluid Mech. 240, 469.Google Scholar
Browand, F. K. & Laufer, J. 1975 Roles of large scale structures in the initial development of circular jets. In Turbulence in Liquids (ed. J. L. Zakin, G. K. Patterson), p. 333. University of Missouri-Rolla.
Broze, G. & Hussain, F. 1991 Spatiotemporal dynamics in transitional jet flows. In Nonlinear Dynamics of Structures (ed. R. Sagdeev, U. Frisch, F. Hussain, S. S. Moiseev & N. S. Grokhin), pp. 391–417. World Scientific.
Broze, G. & Hussain, F. 1994 Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J. Fluid Mech. 263, 93 (referred to herein as BH).Google Scholar
Cohen, J. & Wygnanski, I. 1987 The evolution of instabilities in the axisymmetric jet. Part 2. The flow resulting from the interaction between two waves. J. Fluid Mech. 176, 191.Google Scholar
Coullet, P. & Tresser, J. 1978 Iterations d'endomorphismes et groupe de renormalisation. J. Phys. (Paris) C5, 25.Google Scholar
Cross, M. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851.Google Scholar
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 567.Google Scholar
Drazin, P. G. & Reid, W. H. 1985 Hydrodynamic Stability. Cambridge University Press.
Dubois, M., Rubio, M. A. & Bergé, P. 1983 Experimental evidence of intermittencies associated with a subharmonic bifurcation. Phys. Rev. Lett. 51, 1446.Google Scholar
Feigenbaum, M. J. 1978 Quantitative universality for a class of nonlinear transformations. J. Statist. Phys. 19, 25.Google Scholar
Gersten, K. Kiske, S., Pagendarm, H.-G., Schmitz, O, & Schmucker, A. 1985 Vortex breakdown control by jet injection. Proc. Colloquium on Vortex Breakdown, Feb. 11-12, RWTH Aachen, pp. 101–118.
Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and Groups in Bifurcation Theory, vol. 1. Springer.
Grossmann, S. & Thomae, S. 1977 Invariant distributions and stationary correlation functions of one-dimensional discrete processes. Z. Naturforsch. 32A, 1353.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer. (Corrected second printing, 1986).
Hohenberg, P. C. & Shraiman, B. I. 1989 Chaotic behavior of an extended system. Physica D 37, 109.Google Scholar
Husain, H. S. & Hussain, A. K. M. F. 1989 Subharmonic resonance in a shear layer. In Advances in Turbulence 2 (ed. H. H. Fernholz & H. E. Fielder), pp. 96–101. Springer.
Husain, H. S. & Hussain, F. 1995 Experiments on subharmonic resonance in a shear layer. J. Fluid Mech. 304, 343–372.Google Scholar
Hussain, A. K. M. F. & Zaman, K. M. B. Q. 1978 The free shear layer tone phenomenon and probe interference. J. Fluid Mech. 87, 349.Google Scholar
Hussain, A. K. M. F. & Zaman, K. M. B. Q. 1981 The ‘preferred mode’ of the axisymmetric jet. J. Fluid Mech. 110, 39.Google Scholar
Jensen, M. H., Kadanoff, L. P., Libchaber, A., Procaccia, I. & Stavans, J. 1985 Global universality at the onset of chaos: results of a forced Rayleigh-Bénard experiment. Phys. Rev. Lett. 55, 2798.Google Scholar
Kambe, T. 1969 The stability of an axisymmetric jet with parabolic profile. J. Phys. Soc. Japan 26, 566.Google Scholar
Lessen, M. & Singh, P. J. 1973 The stability of axisymmetric free shear layers. J. Fluid Mech. 60, 443.Google Scholar
Libchaber, A. & Maurer, J. 1980 Une experience de Rayleigh-Bénard de geometrie reduite; multiplication, accrochage et demultiplication de fréquences. J. Phys. (Paris) Lett. 41, C–C51.Google Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistahls unter Berücksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss. 9, 319.Google Scholar
Miksad, R. W., Jones, F. L. & Powers, E. J. 1983 Measurements of nonlinear interactions during natural transition of a symmetric wake. Phys. Fluids 26, 402.Google Scholar
Mollendorf, J. C. & Gebhart, B. 1973 An experimental and numerical study of the viscous instability of a round laminar jet with and without thermal buoyancy for symmetric and asymmetric disturbances. J. Fluid Mech. 61, 367.Google Scholar
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223.Google Scholar
Morris, P. J. 1976 The spatial viscous instability of axisymmetric jets. J. Fluid Mech. 77, 511.Google Scholar
Mullin, T. & Price, T. J. 1989 An experimental observation of chaos arising from the interaction of steady and time-dependent flows. Nature 340, 294.Google Scholar
Muylaert, I. M. 1980 Effect of compressibility on vortex bursting on slender delta wings. VKI Project Rep. 1980-21.Google Scholar
Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom-A attractors near quasiperiodic flow on Tm, m 3. Commun. Math. Phys. 64, 35.Google Scholar
Noack, B. R., Ohle, F. & Eckelmann, H. 1992 Construction and analysis of differential equations from experimental time series of oscillatory systems.. Physica D 56, 389.Google Scholar
Pomeau, Y. & Manneville, P. 1980 Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189.Google Scholar
Ringuet, E., Rozé, C. & Gouesbet, G. 1993 Experimental observation of type-II intermittency in a hydrodynamic system.. Phys. Rev. E 47, 1405.Google Scholar
Ritz, Ch. P. & Powers, E. J. 1986 Estimation of nonlinear transfer functions for fully developed turbulence.. Physica D 20, 320.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167.Google Scholar
Schuster, H. G. 1988 Deterministic Chaos. New York: VCH.
Shtern, V. N. & Hussain, F. 1993 Hysteresis in a swirling jet as a model tornado.. Phys. Fluids A 5, 2183.Google Scholar
Spotar, S. Yu. & Terekhov, V. I. 1987 Two spontaneously alternating regimes of a vortex flow above a plane. J. Appl. Mech. Tech. Phys. N2, 68–70.Google Scholar
Sreenivasan, K. R. 1985 Transition and turbulence in fluid flows and low-dimensional chaos. In Fundamental of fluid mechanics (ed. S. H. Davis & J. L. Lumley), pp. 41–67. Springer.
Van Atta, C. W. & Gharib, M. 1987 Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 174, 113.Google Scholar
Williams-Stuber, K. & Gharib, M. 1990 Transition from order to chaos in the wake of an airfoil. J. Fluid Mech. 213, 29.Google Scholar
Zaman, K. M. B. Q. & Hussain, A. K. M. F. 1980 Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J. Fluid Mech. 101, 449.Google Scholar