Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T02:39:05.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Primary and secondary instabilities in the wake of a cylinder with free ends

Published online by Cambridge University Press:  10 February 1997

Christophe Dauchy ,
Jan Dušek  and
Philippe Fraunié
Christophe Dauchy
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, 12, Avenue du Général Leclerc, 13003 Marseille, Francee-mail: , [email protected]
Jan Dušek
Affiliation:
Institut de Mécanique des Fluides de Strasbourg, 2, rue Boussingault, 67000 Strasbourg, Francee-mail: , [email protected]
Philippe Fraunié
Affiliation:
, LSEET, Université de Toulon et du Var, B. P. 132, 83957 La Garde, Cedex, Francee-mail: , [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The wake of a finite cylinder with free ends and an aspect ratio of 21.4 is simulated in three-dimensions and analysed theoretically. Close to the primary-instability threshold, the flow is shown to settle on a limit cycle with a uniform frequency throughout the flow-field. About 20% above the primary-instability threshold, a secondary instability sets in and the limit cycle becomes unstable. The new attractor of the flow can be identified as a limit T2-torus characterized by two incommensurate frequencies. One of them is shown to evolve continuously from the primary-instability frequency, the other one, about 17 times smaller near the secondary-instability threshold, generates a slow modulation of the oscillations in the wake. The limit cycle and the limit torus are described in terms of their Fourier expansion and the spatial distribution of the most relevant Fourier components is investigated. The theoretical analysis and numerical results given shed some light on the mechanisms underlying a number of known but not satisfactorily explained three-dimensional effects in wakes of finite cylinders such as the ambiguity in the dominant Strouhal frequency, the existence of zones with different frequencies spanwise in the wake, the discreteness of coexisting frequencies observed in the wake as well as the spatial uniformity of the beating period. They moreover explain the Reynolds number variation of these effects and identify the recirculation around the cylinder ends as basically responsible for the onset of the secondary instability. The results are compared to the case of a cylinder with aspect ratio of 10.7 to determine the basic trends in aspect ratio dependence. It is shown that qualitatively the same behaviour is obtained, but that the secondary-instability threshold is shifted significantly upward to about twice the primary-instability threshold. Simulations of the wake of a finite NACA wing with incidence show that the form of the cross-section plays a minor role.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 332 , February 1997 , pp. 295 - 339
Copyright
Copyright © Cambridge University Press 1997

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

Abarbanel, S. S., Don, W. S., Gottlieb, D., Rudy, D. H. & Townsend, J. C. 1991 Secondary frequencies in the wake of a circular cylinder with vortex shedding. J. Fluid Mech. 225, 557.CrossRefGoogle Scholar
Albarede, P. & Provansal, M. 1995 Quasi-periodic cylinder wakes and the Ginzburg-Landau model. J. Fluid Mech. 291, 191.Google Scholar
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215.CrossRefGoogle Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417.CrossRefGoogle Scholar
Berge, P., Pomeau, Y. & Vidal, C. 1988 L'Ordre Dans le Chaos. Hermann.Google Scholar
Berger, E. 1964 Bestimmung der hydrodynamischen Groessen einer Karmanschen Wirbelstarsse aus Hitzdrahtmessungen bei kleinen Reynoldsschen Zahlen. Z. Flugwiss. 12, 41.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.CrossRefGoogle Scholar
Brede, M., Eckelmann, H., Konig, M. & Noack, B. R. 1994 Discrete shedding modes of the cylinder wake in a jet with homogeneous core. Phys. Fluids 6, 2711.Google Scholar
Carte, G., Dusek, J. & Fraunie, P. 1995 A spectral time discretization for flows with dominant periodicity. J. Comput. Phys. 120, 171.CrossRefGoogle Scholar
Carte, G., Dusek, J. & Fraunie. P. 1996 Numerical simulation of the mechanisms governing the onset of the Benard-von Karman instability. Intl J. Numer. Meth. Fluids, 23, 753.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. Sixth Intl Symp. on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines and Propellers, University Notre Dame, USA, Sept. 15-19, 1991, Springer.Google Scholar
Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Ann. Rev. Fluid Mech 23, 341.CrossRefGoogle Scholar
Dauchy, C. 1995 Etude numerique d'une instabilite secondaire dans le sillage de cylindres finis. PhD thesis, IRPHE, Universite d'Aix-Marseille II.Google Scholar
Dusek, J. 1996 Spatial structure of the Benard-von Karman instability. Eur. J. Mech. B Fluids. 15, 619.Google Scholar
Dusek, J. & Fraunie, P. 1993 Validation and error estimate of a spectral element discretization of a cylinder wake. In Numerical Methods in Laminar and Turbulent Flows'93, vol. VIII, Part 1 (ed. Taylor, C.). Pineridge Press.Google Scholar
Dusek, J., Fraunie, P. & SEROR, S. 1994a Mise en evidence du doublement de periode dans ie sillage d'une aile NACA a la deuxieme bifurcation de Hopf. C. R. Acad. Sci. Paris 319, II, 1271.Google Scholar
Dusek, J., LE GAL, P. & Fraunie, P. 19946 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59.CrossRefGoogle Scholar
Eisenlohr, H. & Eckelmann, H. 1989 Vortex splitting and its consequence in the vortex street wake of cylinders at low Reynolds numbers. Phys. Fluids A 1, 189.Google Scholar
Gaster, M. 1969 Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 38, 565.CrossRefGoogle Scholar
Gaster, M. 1971 Vortex shedding from circular cylinders at low Reynolds numbers J. Fluid Mech. 46, 749.Google Scholar
Gerich, D. & Eckelmann, H. 1982 Influence of end plates and free ends on the shedding frequency of circular cylinders. J. Fluid Mech. 122, 109.CrossRefGoogle Scholar
Goujon-Durand, S., Jenffer, P. & Wesfreid, J. E. 1994 Downstream evolution of the Benard von Karman instability. Phys. Rev. £ 50, 1.Google Scholar
Hammache, M. & Gharib, M. 1989 A novel method to promote parallel vortex shedding in the wake of circular cylinders. Phys. Fluids A 1, 1611.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.CrossRefGoogle Scholar
Jayaweera, K. O. L. F. & Mason, B. J. 1965 The behaviour of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech. 22, 709.CrossRefGoogle Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1989 Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199, 441.CrossRefGoogle 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.CrossRefGoogle Scholar
Konig, M. 1993 Experimented Untersuchung des dreidimensionalen Nachlaufs zylindrischer Korper bei kleinen Reynoldszahlen. Mitteilungen aus dem Max-Planck-Institut fur Stromungsforschung 111.Google Scholar
Konig, M., Eisenlohr, H. & Eckelmann, H. 1990 The fine structure in the Strouhal-Reynolds number relationship of the laminar wake of a circular cylinder. Phys. Fluids A 2, 1607.CrossRefGoogle Scholar
Konig, M., Eisenlohr, H. & Eckelmann, H. 1992 Visualization of the spanwise cellular structure of the laminar wake of wall-bounded circular cylinders. Phys. Fluids A 4, 869.CrossRefGoogle Scholar
Konig, M., Noack, B. R. & Eckelmann, H. 1993 Discrete shedding modes in the von Karman vortex street. Phys. Fluids A 5, 1846.CrossRefGoogle Scholar
Korczak, K. Z. & Patera, A. T. 1986 An isoparametric spectral element method for solution of the Navier-Stokes equation in complex geometry. J. Comput. Phys. 62, 361.CrossRefGoogle Scholar
Landau, L. D. & Lifschitz, F M. 1959 Fluid Mechanics, Course of Theoretical Physics, Vol 6. Pergamon Press.Google Scholar
Le Gal, P., Chauve, M. P., Lima, R. & Rezende, J. 1990 Coupled wakes behind two circular cylinders. Phys. Rev. A 41, 4566.Google Scholar
Leweke, T. & Provansal, M. 1995 The flow behind rings: bluff body wakes without end effects. J. Fluid Mech. 288, 265.CrossRefGoogle Scholar
Mathis, C. 1983 Proprietes de vitesse transverses dans l'ecoulement de Benard von Karman aux faibles nombres de Reynolds. These, Universite Aix-Marseille III.Google Scholar
Mathis, C., Provansal, M. & Boyer, L. 1987 The Benard-von Karman instability: transient and forced regimes. J. Fluid Mech. 182, 1.Google Scholar
Mattingly, G .E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid. Mech. 51, 233.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279.CrossRefGoogle Scholar
Newhouse, S., Ruelle, D. & TAKENS, F. 1978 Occurrence of strange axiom Aattractors near quasi periodic flows. Commun. Math. Phys. 64, 35.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297.CrossRefGoogle Scholar
Noack, B. R., K5NIG, M. & Eckelmann, H. 1993 Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5, 1279.Google Scholar
Olinger, D. J. 1990 Universality in the transition to chaos in open fluid systems. PhD thesis, Yale University.Google Scholar
Olinger, D. J. & Sreenivasan, K. R. 1988 Non linear dynamics of the wake of an oscillating cylinder. Phys. Rev. Lett. 60, 797.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
Peschard, I. 1995 De l'oscillateur aux sillages couples. These, Universite d'Aix-Marseille II.Google Scholar
Roshko, A. 1954 On the development of turbulent wakes from vortex streets. NACA Rep. 1191.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167.Google Scholar
Sirovich, L. 1985 The Karman vortex trail and flow behind a circular cylinder. Phys. Fluids 28, 2723.Google Scholar
Sreenivasan, K. R. 1985 Transition and turbulence in fluid flows and low-dimensional chaos. In Frontiers in Fluid Mechanics (ed. Davis, S. H. & Lumley, J. L.), pp. 41–67. Springer.Google Scholar
Sreenivasan, K. R., Strykowski, P. J. & OLINGER, D. J. 1987 Hopf bifurcation, Landau equation, and vortex shedding behind circular cylinders. 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.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
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6, 547.Google Scholar
Tritton, D. J. 1971 A note on vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 45, 203.CrossRefGoogle Scholar
Van Atta, C. & Gharib, M. 1987 Ordered and chaotic vortex streets behind cylinders at low Reynolds numbers. J. Fluid Mech. 174, 113.Google Scholar
Wesfreid, J. E., Goujon-Durand, S. & Zielinska, B. J. A. 1995 Global mode behaviour of the streamwise velocity in wakes. J. Phys. II Paris, 6, 1343.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
Williamson, C. H. K. 1992 The natural and forced formation of spot-like ‘vortex dislocations’ in the transition of a wake. J. Fluid Mech. 243, 393.Google Scholar
Williamson, C. H. K. & PRASAD, A. 1993 A new mechanism for oblique wave resonance in the ‘natural’ far wake. J. Fluid Mech. 256, 269.CrossRefGoogle Scholar
Zhang, H.-Q., FEY, U., NOACK, B. R., Konig, M. & Eckelmann, H. 1995 On the transition of the cylinder wake, Phys. Fluids 7, 779.Google Scholar