Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T06:58:29.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors

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

Conclusive experimental evidence is presented for the existence of a low-dimensional temporal dynamical system in an open flow, namely the near field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 × 104 [Lt ] ReD × 9.1 × 104) with a top-hat profile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters – forcing frequency StD (≡fexD/Ue, where fez is the excitation frequency, D is the jet exit diameter and Ue is the exit velocity) and forcing amplitude af (≡ u’f/Ue, where u’f is the jet exit r.m.s. longitudinal velocity fluctuation at the excitation frequency) – were varied over the ranges 10-4 < af < 0.3 and 0.3 < StD < 3.0 in order to construct a phase diagram. Periodic and chaotic states were found over large domains of the parameter space. The periodic attractors correspond to stable pairing (SP) and stable double pairing (SDP) of rolled-up vortices. One chaotic attractor, near SP in the parameter space, results from nearly periodic modulations of pairing (NPMP) of vortices. At large scales (i.e. approximately the size of the attractor) in phase space, NPMP exhibits approximately quasi-periodic behaviour, including modulation sidebands around ½fex in u-spectra, large closed loops in its Poincaré sections, correlation dimension v ∼ 2 and largest Lyapunov exponent λ1 ∼ 0. But investigations at smaller scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown by v > 2 and λ1 > 0. The other chaotic attractor, near SDP, results from nearly periodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 [Lt ] v [Lt ] 3 and the largest Lyapunov exponent 0.2 [Lt ] λ1 [Lt ] 0.7 bits per orbit (depending on measurement locations in physical and parameter spaces).

A definition that distinguishes between physically and dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physically open flow, even one that is apparently dynamically open due to convective instability, can exhibit dynamically closed behaviour as a result of feedback. A conceptual model for transitional jets is proposed based on twodimensional instabilities, subharmonic resonance and feedback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental–subharmonic phase difference ϕ – a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pairings to ϕ, the phase diagram, and the documented periodic and chaotic attractors demonstrate the validity of the proposed conceptual model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 263 , 25 March 1994 , pp. 93 - 132
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.)

References

Aronson, I. S., Gaponov-Grekhov, A. V. & Rabinovich, M. I. 1988 Physica 33D, 1.
Aubry, N. Holmes, P., Lumley, J. L. & Stone, E. 1988 J. Fluid Mech. 192, 115.
Bergé, P., Dubois, M., Manneville, P. & Pomeau, Y. 1980 J. Phys. (Paris) Lett. 41, L344.
Berger, W. D. 1993 Visualization study of some vortex interactions in a circular jet. Senior Honors thesis, University of Houston.
Biringen, S. & Peltier, L. J. 1990 Phys. Fluids A 2, 754.
Bonetti, M. & Boon, J.-P. 1989 Phys. Rev A 40, 3322.
Brandstäter, A. & Swinney, H. L. 1987 Phys. Rev. A 35, 2207.
Bridges, J. E. 1990 Application of coherent structure vortex sound theories to jet noise. PhD dissertation, University of Houston.
Bridges, J. E. & Hussain, F. 1992 J. Fluid Mech. 240, 469.
Brown, G. L. & Roshko, A. 1974 J. Fluid Mech. 64, 775.
Broze, G. 1992 Chaos in an ‘open’ flow: experiments in transitional jets. PhD dissertation, University of Houston.
Broze, G. & Hussain, F. 1991 In Nonlinear Dynamics of Structures (ed. R. Sagdeev, U. Frisch, F. Hussain, S. Moiseev & N. Erokhin), pp. 391–417. World Scientific.
Buell, J. C. & Huerre, P. 1988 In Proc. Summer Institute of the Center for Turbulence Research, NASA Ames—Stanford University, pp. 19–27.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Phys. Rev. Lett. 60, 25.
Corcos, G. M. & Sherman, F. S. 1984 J. Fluid Mech. 139, 29.
Crow, S. C. & Champagne, F. H. 1971 J. Fluid Mech. 48, 567.
Deissler, R. J. 1985 J. Statist. Phys. 40, 371.
Deissler, R. J. 1989 J. Statist. Phys. 54, 1459.
Deissler, R. J. & Kaneko, K. 1987 Phys. Lett. A 119, 397.
Dimotakis, P. E. & Brown, G. L. 1976 J. Fluid Mech. 78, 535.
Fiedler, H. E. 1988 Prog. Aerospace Sci. 25, 231.
Fraser, A. & Swinney, H. L. 1986 Phys. Rev. A 33, 1134.
Freymuth, P. 1966 J. Fluid Mech. 25, 683.
Glauser, M., Zheng, X. & Doering, C. R. 1991 Bull. Am. Phys. Soc. 36, 2621.
Grassberger, P. & Procaccia, I. 1983 Physica 9D, 189.
Grinstein, F. F., Oran, E. S. & Boris, J. P. 1990 Phys. Rev. Lett. 64, 870.
Guckenheimer, J. 1986 A. Rev. Fluid Mech. 18, 15.
Huerre, P. & Monkewitz, P. A. 1985 J. Fluid Mech. 159, 151.
Huerre, P. & Monkewitz, P. A. 1990 A. Rev. Fluid Mech. 22, 472.
Husain, H. S. & Hussain, A. K. M. F. 1986 Bull. Am. Phys. Soc. 31, 1696.
Husain, H. S. & Hussain, A. K. M. F. 1989 In Advances in Turbulence 2 (ed. H. Fernholz & H. Fiedler), pp. 96–101. Springer.
Hussain, A. K. M. F. 1986 J. Fluid Mech. 173, 303.
Hussain, A. K. M. F., Husain, H. S., Zaman, K. B. M. Q., Tso, J., Hayakawa, M., Takaki, R. & Hasan, M. A. Z. 1986 AIAA Paper 86-0235.
Hussain, A. K. M. F. & Zaman, K. M. B. Q. 1978 J. Fluid Mech. 87, 349.
Hussain, A. K. M. F. & Zaman, K. M. B. Q. 1980 J. Fluid Mech. 101, 493.
Hussain, A. K. M. F. & Zaman, K. M. B. Q. 1981 J. Fluid Mech. 110, 39.
Jenkinson, J. P. & Hussain, A. K. M. F. 1987 Bull. Am. Phys. Soc. 32, 2026.
Keefe, L., Moin, P. & Kim, J. 1992 J. Fluid Mech. 242, 1.
Keefe, L. R. 1987 Bull. Am. Phys. Soc. 32, 2070.
Kelly, R. E. 1967 J. Fluid Mech. 27, 657.
Kibens, V. 1980 AIAA J. 18, 434.
Kline, S. J., Reynolds, W. D., Schraub, F. A. & Runstadler, P. W. 1967 J. Fluid Mech. 30, 741.
Lanford, O. E. 1982 A. Rev. Fluid Mech. 14, 347.
Laufer, J. & Monkewitz, P. 1980 AIAA Paper 80-0962.
Libchaber, A. & Maurer, J. 1980 J. Phys. (Paris) Lett. 41, C3–51.
Lumley, J. L. 1981 In Transition and Turbulence (ed. R. E. Meyer), p. 215. Academic.
Mankbadi, R. R. 1985 J. Fluid Mech. 160, 385.
Michalke, A. 1965 J. Fluid Mech. 23, 521.
Michalke, A. 1971 Z. Flugwiss. 19, 159.
Mitschke, F., Möller, M. & Lange, W. 1988 Phys. Rev. A 37, 4518.
Monkewitz, P. A. 1988 J. Fluid Mech. 188, 223.
Monkovin, M. V. 1988 AIAA paper 88-3675.
Patnaik, P. C., Sherman, F. S. & Corcos, G. M. 1976. J. Fluid Mech. 73, 216.
Pierrehumbert, R. T. & Widnall, S. E. 1981 J. Fluid Mech. 102, 301.
Rohling, T., Ghia, K. N., Osswald, G. A. & Ghia, U. 1990 Bull. Am. Phys. Soc. 35, 2229.
Ruelle, D. 1990 Proc. R. Soc. Lond. A 427, 241.
Ruelle, D. 1991 In New Perspectives in Turbulence (ed. L. Sirovich), pp. 123–138. Springer.
Sreenivasan, K. R. 1985 In Fundamentals of Fluid Mechanics (ed. S. H. Davis & J. L. Lumley), pp. 41–67. Springer.
Sreenivasan, K. R. 1986 In Dimensions and Entropies in Chaotic Systems (ed. G. Mayer-Kress), pp. 222–230. Springer.
Theiler, J. 1986 Phys. Rev. A 34, 2427.
Van Atta, C. W. & Gharib, M. 1987 J. Fluid Mech. 174, 113.
Vastano, J. & Pulliam, T. 1989 In Proc. ASME-ASCE Forum on Chaotic Dynamics, UCSD, La Jolla, CA, July 10–12, 1989.
Virk, D. P. S. 1989 Numerical study of feedback and subharmonic resonance in free shear layers. MS thesis, University of Houston.
Williams-Stuber, K. & Gharib, M. 1990 J. Fluid Mech. 213, 29.
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. 1985 Physica 16D, 285.
Yokuda, S. & Ramaprian, B. R. 1990 Phys. Fluids A 2, 784.
Zaman, K. M. B. Q. & Hussain, A. K. M. F. 1980 J. Fluid Mech. 101, 449.