Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T15:40:02.533Z Has data issue: false hasContentIssue false
  • English
  • Français

Measurements of spatiotemporal dynamics in a forced plane mixing layer

Published online by Cambridge University Press:  26 April 2006

Satish Narayanan  and
Fazle Hussain
Satish Narayanan
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

We present an approach combining temporal dynamical systems methods with newly proposed spatial coupling measures – namely, coherence and cross-bicoherence – to identify and quantitatively describe low-dimensional dynamics in transitional open flows. The approach is used to describe a forced mixing layer as a low-dimensional temporal dynamical system and interpret its transitional vortex dynamics.

Experiments were performed in an initially laminar plane mixing layer inside an anechoic chamber using forcing of the fundamental instability only; the forcing frequency and amplitude are used as control parameters. Dynamical invariants calculated show that vortex roll-up and the feedback-driven first two pairing dynamics are well-described by one periodic and at least two low-dimensional chaotic attractors; a phase diagram delineating such dynamical states in the control parameter space is presented. The large spatial extents of these feedback-sustained states (verified using coherence and cross-bicoherence), spanning many instability wavelengths downstream, indicate spatial coupling; feedback has also been experimentally verified. At a fixed forcing frequency, as the forcing amplitude is decreased, the spatially coupled, periodic second pairing dynamics becomes chaotic and spatiotemporal (inferred from decay of coherence and cross-bicoherence); the dynamics in the domain that includes the first pairing, however, remains temporal. This loss of spatial coupling is accompanied by a sudden increase in the attractor dimension, and suggests spatiotemporal chaos. The combination of dynamical systems theory and spatial measures seems to be a promising approach to probe spatiotemporal dynamics in other open flows as well.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 320 , 10 August 1996 , pp. 71 - 115
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

Bendat, J. S. & Piersol, A. G. 1986 Random Data Analysis and Measurement Procedures, 2nd edn. Wiley-Interscience.
Bonetti, M. & Boon, J.-P. 1989 Chaotic dynamics in an open flow: the excited jet. Phys. Rev. A 40, 3322.Google Scholar
Bradshaw, P. 1964 Wind tunnel screens: flow instability and its effect on aerofoil boundary layers. J.R. Aeronaut. Soc. 68, 198.Google Scholar
Brandstäter, A., Swift, J., Swinney, H. L., Wolf, A., Farmer, J. D. & Jen, E. 1983 Low-dimensional chaos in a hydrodynamic system. Phys. Rev. Lett. 51, 1442.Google Scholar
Bridges, J. E. 1990 Application of coherent structure and vortex sound theories to jet noise. PhD dissertation, University of Houston.
Brown, G. B. 1937 The vortex motion causing edge tones. Proc. Phys. Soc. 49, 493.Google Scholar
Broze, G. 1992 Chaos in an ‘open’ flow: experiments in transitional jets. PhD dissertation, University of Houston.
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
Broze, G. & Hussain, F. 1996 Transition to chaos in a forced jet: intermittency, tangent bifurcations and hysteresis. J. Fluid Mech. 311, 37.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851.Google Scholar
Deissler, R. J. & Kaneko, K. 1987 Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems. Phys. Lett. 119, 397.Google Scholar
Dimotakis, P. E. & Brown, G. L. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78, 535.Google Scholar
Dubois, M. 1982 Experimental aspects of the transition to turbulence in Rayleigh—Bénard convection. Stability of Thermodynamic Systems. Lecture Notes in Physics, vol. 164, p. 117.
Fraser, A. & Swinney, H. L. 1986 Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134.Google Scholar
Freymuth, P. 1966 On transition in a separated laminar boundary layer. J. Fluid Mech. 25, 683.Google Scholar
Grassberger, P. & Procaccia, I. 1983 Measuring the strangeness of strange attractors. Physica D 9, 189.Google Scholar
Grinstein, F. F., Oran, E. S. & Boris, J. P. 1991 Pressure field, feedback, and global instabilities of subsonic spatially developing mixing layers. Phys. Fluids A 3, 2401.Google Scholar
Heutmaker, M. S. & Gollub, J. P. 1987 Wave-vector field of convective flow patterns. Phys. Rev. A 35, 242.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365.Google Scholar
Hohenberg, P. C. & Shraiman, B. I. 1989 Chaotic behavior of an extended system. Physica D 37, 109.Google Scholar
Huerre, P. 1987 Spatio-temporal instabilities in closed and open flows. In Instabilities and Nonequilibrium Structures (ed. E. Tirapegui & D. Villaroel), p. 141. Reidel.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473 (herein referred to as HM).Google Scholar
Husain, H. S. & Hussain, F. 1989 Subharmonic resonance in a shear layer. In Advances in Turbulence 2 (ed. H. H. Fernholz & H. E. Fiedler), p. 96, Springer; see also J. Fluid Mech. 304, 343 (1995).
Hussain, F. 1983 Coherent structures — reality and myth. Phys. Fluids 26, 2816.Google Scholar
Hussain, F., Husain, H. S., Zaman, K. B. M. Q., Tso, J., Hayakawa, M., Takaki, R. & Hasan, M. A. Z. 1986 Free shear flows: organized structures and effects of excitation. AIAA paper 86-0235.
Hussain, F. & Zaman, K. B. M. Q. 1978 The free shear layer tone phenomenon and probe interference. J. Fluid Mech. 87, 349.Google Scholar
Jeong, J. & Hussain, F. 1992 Coherent structure near the wall in a turbulent boundary layer. The Fifth Asian Congress of Fluid Mechanics, Aug. 10-14, Taejon, Korea, vol. 21, p. 1262.
Kaneko, K. 1984 Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice. Prog. Theor. Phys. 72, 202.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657.Google Scholar
Laws, E. M. & Livesey, J. L. 1978 Flow through screens. Ann. Rev. Fluid Mech. 10, 247.Google Scholar
Mathis, C., Provansal, M. & Boyer, L. 1984 The Bénard—von Kármán instability: an experimental study near the threshold. J. Phys. Lett. Paris 45, L483.Google Scholar
Mayer-Kress, G. & Kurz, T. 1987 Dimension densities for turbulent systems with spatially decaying correlation functions. Complex Systems 1, 821.Google Scholar
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521.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
Monkewitz, P. A. 1988 Subharmonic resonance, pairing and shredding in the mixing layer. J. Fluid Mech. 188, 223.Google Scholar
Monkewitz, P. A., Bechert, D. W., Barsikow, B. & Lehmann, B. 1990 Self-excited oscillations and mixing in a heated round jet. J. Fluid Mech. 213, 611.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear stability of weakly non-parallel shear flows. J. Fluid Mech. 251, 1.Google Scholar
Oster, D. & Wygnanski, I. 1982 The forced mixing layer between parallel streams. J. Fluid Mech. 123, 91.Google Scholar
Ott, E., Grebogi, C. & Yorke, J. A. 1990 Controlling chaos. Phys. Rev. Lett. 64, 1196.Google Scholar
Pomeau, Y. 1985 Méchanique des fluides — measure de la densité d'information en turbulence. C. R. Acad. Sci. Paris, 300, Série II(7), 239.Google Scholar
Rasmussen, D. R. & Bohr, T. 1987 Temporal chaos and spatial disorder. Phys. Lett. A 125, 107.Google Scholar
Reisenthel, P. 1988 Hybrid instability in an axisymmetric jet with enhanced feedback. PhD dissertation, Illinois Institute of Technology.
Ritz, Ch.P. & Powers, E. J. 1986 Estimation of nonlinear transfer functions for fully developed turbulence. Physica D 20, 320.Google Scholar
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations and their control. J. Fluid Mech. 271, 17.Google Scholar
Theiler, J., Eubank, S., Longtin, A., Galdrikian, B. & Farmer, J. D. 1992 Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77.Google Scholar
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
William-Stuber, K. & Gharib, M. 1990 Transition from order to chaos in the wake of an airfoil. J. Fluid Mech. 213, 29.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing, the mechanism of turbulent mixing layer growth. J. Fluid Mech. 63, 237.Google Scholar
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica D 16, 285.Google Scholar
Zaman, K. B. M. Q. & Hussain, F. 1980 Vortex pairing in a circular jet under controlled excitation. Part 1. General jet response. J. Fluid Mech. 101, 449.Google Scholar