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A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake

Published online by Cambridge University Press:  26 April 2006

Jan Dušek
Affiliation:
Institut de Mécanique Statistique de la Turbulence, 12, Av. Général Leclerc, 13003 Marseille, France
Patrice Le Gal
Affiliation:
Institut de Mécanique Statistique de la Turbulence, 12, Av. Général Leclerc, 13003 Marseille, France
Philippe Fraunié
Affiliation:
Institut de Mécanique Statistique de la Turbulence, 12, Av. Général Leclerc, 13003 Marseille, France Present address: LSEET, Université de Toulon et du Var, BP 132, 83954 La Garde, Cédex, France.

Abstract

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.

Type
Research Article
Copyright
© 1994 Cambridge University Press

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References

Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359.Google Scholar
Benney, D. J. 1960 A non-linear theory for oscillations in a parallel flow. J. Fluid Mech. 10, 209.Google Scholar
Braza, M., Chassaing, P. & Ha Minh, H. 1986 Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79.Google Scholar
Carte, G., Fraunie, P. & Dussouillez, P. 1991 An innovative algorithm for periodic flows calculation using a parallel architecture. Some Applications for unsteady aerodynamics. In Sixth Intl Symp. on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines and Propellers, Univ. Notre Dame, USA, Sept. 15–19, 1991 (ed. H. M. Atassi). Springer.
Croquette, V. & Williams, H. 1989 Non-linear waves of the oscillatory instability in finite convective rolls. Physica D 37, 300.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11, 723.Google Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473.Google Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1989 Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199, 441.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1992 The three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech. 238, 1.Google Scholar
Kolodner, P. & Williams, H. 1990 Dispersive chaos. In Proc. NATO Advanced Research Workshop on Nonlinear Evolution of Spatio-temporal structures in Dissipative Continuous Systems (ed. F. H. Busse & L. Kramer). NATO Series B2.225, p. 73. Plenum.
Landau, L. D. & Lifschitz, F. M. 1959 Fluid Mechanics, Course of Theoretical Physics, vol. 6. Pergamon.
Le Gal, P. 1992 Complex demodulation applied to the transition to turbulence of the flow over a rotating disk. Phys. Fluids A 4, 2523.Google Scholar
Li, J., Sun, J. & Roux, B. 1992 Numerical study of an oscillating fluid cylinder in uniform flow and in the wake of an upstream cylinder. J. Fluid Mech. 237, 457.Google Scholar
Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic.
Mathis, C. 1983 Propriétés de vitesse transverses dans l’écoulement de Bénard von Kármán aux faibles nombres de Reynolds. Thèse, Université Aix-Marseille.
Mathis, C., Provansal, M. & Boyer, L. 1987 The Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54, 468.Google Scholar
Raghu, S. & Monkewitz, P. A. 1991 The bifurcation of a hot round jet to limit-cycle oscillations. Phys. Fluids A 3, 501.Google Scholar
Sreenivasan, K. R., Strykowski, P. J. & Olinger, D. J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. In Proc. Forum on Unsteady Flow Separation, ASME Applied Mechanics, Bio engineering and Fluid Engineering Conference, Cincinnati, Ohio, June 11–17, 1987. ASME FED, Vol. 52.
Stewartson, K. & Stuart, J. T. 1971 A non-linear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529.Google Scholar
Strykowski, P. J. & Sreenivasan, K. R. 1990 On the formation and suppression of vortex shedding at low Reynolds numbers. J. Fluid Mech. 218, 71.Google Scholar
Williamson, C. H. K. 1989 Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 206, 579.Google Scholar