Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-19T18:33:31.375Z Has data issue: false hasContentIssue false
  • English
  • Français

The flow behind rings: bluff body wakes without end effects

Published online by Cambridge University Press:  26 April 2006

T. Leweke  and
M. Provansal
T. Leweke
Affiliation:
Laboratoire de Recherche en Combustion, URA 1117 CNRS/Université de Provence, Faculté de Saint-Jérôme, Service 252, F-13397 Marseille Cédex 20, France
M. Provansal
Affiliation:
Laboratoire de Recherche en Combustion, URA 1117 CNRS/Université de Provence, Faculté de Saint-Jérôme, Service 252, F-13397 Marseille Cédex 20, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent studies have demonstrated the strong influence of end effects on low-Reynoldsnumber bluff body wakes, and a number of questions remain concerning the intrinsic nature of three-dimensional phenomena in two-dimensional configurations. Some of them are answered by the present study which investigates the wake of bluff rings (i.e. bodies without ends) both experimentally and by application of the phenomenological Ginzburg–Landau model. The model turns out to be very accurate in describing qualitative and quantitative observations in a large Reynolds number interval. The experimental study of the periodic vortex shedding regime shows the existence of discrete shedding modes, in which the wake takes the form of parallel vortex rings or ‘oblique’ helical vortices, depending on initial conditions. The Strouhal number is found to decrease with growing body curvature, and a global expression for the Strouhal–Reynolds number relation, including curvature and shedding angle, is proposed, which is consistent with previous straight cylinder results. A secondary instability of the helical modes at low Reynolds numbers is discovered, and a detailed comparison with the Ginzburg–Landau model identifies it as the Eckhaus modulational instability of the spanwise structure of the near-wake formation region. It is independent of curvature and its clear observation in straight cylinder wakes is inhibited by end effects.

The dynamical model is extended to higher Reynolds numbers by introducing variable parameters. In this way the instability of periodic vortex shedding which marks the beginning of the transition range is characterized as the Benjamin–Feir instability of the coupled oscillation of the near wake. It is independent of the shear layer transition to turbulence, which is known to occur at higher Reynolds numbers. The unusual shape of the Strouhal curve in this flow regime, including the discontinuity at the transition point, is qualitatively reproduced by the Ginzburg–Landau model. End effects in finite cylinder wakes are found to cause important changes in the transition behaviour also: they create a second Strouhal discontinuity, which is not observed in the present ring wake experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 265 - 310
Copyright
© 1995 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

Albarède, P. 1991 Self-organization in three-dimensional wakes of bluff bodies. PhD thesis, Université de Provence, Marseille, France.
Albarède, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding patterns and ‘chevrons’ in cylinder wakes. Phys. Fluids A 4, 744.Google Scholar
Albarède, P. & Provansal, M. 1995 Quasi-periodic cylinder wakes and the Ginzburg–Landau model. J. Fluid Mech. (To appear).Google Scholar
Albarède, P., Provansal, M. & Boyer, L. 1990 The Ginzburg–Landau equation as a model for the three-dimensional wake of an elongated bluff body. C. R. Acad. Sci. Paris 310 (II), 459.Google Scholar
Bearman, P. W. & Takamoto, M. 1988 Vortex shedding behind rings and disks. Fluid Dyn. Res. 3, 214.Google Scholar
Berger, E. & Wille, R. 1972 Periodic flow phenomena. Ann. Rev. Fluid Mech. 4, 313.Google Scholar
Bloor, S. 1964 The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech. 19, 290.Google Scholar
Brede, M., Eckelmann, H., König, M. & Noack, B. R. 1994 Discrete shedding modes of the cylinder wake in a jet with a homogeneous core. Phys. Fluids 6, 2711.Google Scholar
Chiffaudel, A. 1992 Non-linear stability analysis of two-dimensional patterns in the wake of a circular cylinder. Europhys. Lett. 18, 589.Google Scholar
Coutanceau, M. & Defaye, J.-R. 1991 Circular cylinder wake configurations: a flow visualisation survey. Appl. Mech. Rev. 44, 255.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851.Google Scholar
Ehrhardt, G. 1979 Stabilität zweireihiger Straßen geradliniger und kreisförmiger Wirbel. Fortschrittsber. der VDI-Z., Reihe 7, no. 49.Google Scholar
Eisenlohr, H. & Eckelmann, H. 1989 Vortex splitting and its consequences in the vortex street wake of cylinders at low Reynolds number. Phys. Fluids A 1, 189.Google Scholar
Gaster, M. 1969 Vortex shedding from slender cones at low Reynolds numbers, J. Fluid Mech. 38, 565.Google 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 Influnce of end plates and free ends on the shedding frequencies of circular cylinders. J. Fluid Mech. 122, 109.Google Scholar
Gerrard, J. H. 1978 The wakes of cylindrical bluff bodies at low Reynolds number. Phil. Trans. R. Soc. Lond. A 288, 351.Google Scholar
Goujon-Durand, S., Jenffer, P. & Wesfreid, J. E. 1994 Downstream evolution of the Bénard–von Kármán instability. Phys. Rev. E 50, 308.Google Scholar
Hama, F. R. 1957 Three-dimensional vortex pattern behind a circular cylinder. J. Aero. Sci. 24, 156.Google Scholar
Hammache, M. & Gharib, M. 1989 A novel method to promote parallel shedding in the wake of circular cylinders. Phys. Fluids A 1, 1611.Google Scholar
Hammache, M. & Gharib, M. 1991 An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mech. 232, 567.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473.Google Scholar
Janiaud, B., Pumir, A., Bensimon, D., Croquette, V., Richter, H. & Kramer, L. 1992 The Eckhaus instability for traveling waves. Physica D 55, 269.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 Three-dimensional dynamics and transition to no turbulence in the wake of bluff objects. J. Fluid Mech. 238, 1.Google Scholar
König, M. 1993 Experimentelle Untersuchung des dreidimensionalen Nachlaufs zylindrischer Körper bei kleinen Reynoldszahlen. Mitt. Max-Planck-Institut für Strömungsforschung 111.Google Scholar
König, 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.Google Scholar
König, M., Noack, B. R. & Eckelmann, H. 1993 Discrete shedding modes in the von Kármán vortex street. Phys. Fluids A 5, 1846.Google Scholar
Kramer, L. & Zimmermann, W. 1985 On the Eckhaus instability for spatially periodic patterns. Physica 16D, 221.Google Scholar
Kuramoto, Y. 1984 Chemical Oscillations, Waves, and Turbulence, Appendix A. Springer.
Leweke, T. 1994 Experimental study and modelling of the wake of a ring at low Reynolds numbers. PhD thesis, Université de Provence, Marseille, France.
Leweke, T. & Provansal, M. 1994a Model for the transition in bluff body wakes. Phys. Rev. Lett. 72 3174.Google Scholar
Leweke, T. & Provansal, M. 1994b Determination of the parameters of the Ginzburg–Landau wake model from experiments on a bluff ring. Europhys. Lett. 27, 655.Google Scholar
Leweke, T., Provansal, M. & Boyer, L. 1993a Three-dimensional wake of a torus and its modelling by the Ginzburg–Landau equation. C. R. Acad. Sci. Paris 316 (II), 287.Google Scholar
Leweke, T., Provansal, M. & Boyer, L. 1993b Stability of vortex shedding modes in the wake of a ring at low Reynolds numbers. Phys. Rev. Lett. 71, 3469.Google Scholar
Lowe, M. & Gollub, J. P. 1985 Pattern selection near the onset of convection: the Eckhaus instability. Phys. Rev. Lett. 55, 2575.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
Miller, G. D. & Williamson, C. H. K. 1994 Control of three-dimensional phase dynamics in a cylinder wake. Exps. Fluids 18, 26.Google Scholar
Monson, D. R. 1965 Experimental drag characteristics of tori, ducted-spheres, and other shapes at low Reynolds numbers. Masters thesis, University of Minnesota.
Monson, D. R. 1981 The effect of transverse curvature on the drag and vortex shedding of elongated bluff bodies at low Reynolds numbers. ASME Paper 81-WA/FF-4.
Monson, D. R. 1983 The effect of transverse curvature on the drag and vortex shedding of elongated bluff bodies at low Reynolds numbers. Trans. ASME I: J. Fluids Engng. 105, 308.Google Scholar
Noack, B. R., König, M. & Eckelmann, H. 1993 Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5, 1279.Google Scholar
Norberg, C. 1994 An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287.Google Scholar
Park, D. S. & Redekopp, L. G. 1992 A model for pattern selection in wake flows. Phys. Fluids A 4, 1697.Google Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1.Google Scholar
Rivet, J.-P. 1991 Spontaneous symmetry-breaking in the 3-D wake of a bluff cylinder, simulated by the lattice gas method. C. R. Acad. Sci. Paris 313 (II), 151.Google Scholar
Roshko, A. 1953 On the development of turbulent wakes from vortex streets. NACA Tech. Note 2913.Google Scholar
Saffmann, P. G. 1992 Vortex Dynamics, chap. 10. Cambridge University Press.
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of two–dimensional bluff bodies and their control. J. Fluid Mech. 271, 17.Google Scholar
Shraiman, B. I., Pumir, A., Saarlos, W. Van, Hohenberg, P. C., Chaté, H. & Holen, M. 1992 Spatiotemporal chaos in the one-dimensional complex Ginzburg–Landau equation. Physica D 57, 241.Google Scholar
Sreenivasan, K. R. 1985 Transition and turbulence in fluid flows and low-dimensional chaos. In Frontiers in Fluid Mechanics (ed. S. H. Davis & J. L. Lumley), pp. 41–66. Springer.
Sreenivasan, K. R., Strykowski, P. J. & Ohlinger, D. J. 1986 Hopf bifurcation, Landau equation and vortex shedding behind circular cylinders. In Proc. Forum on Unsteady Flow Separation (ed. K. N. Ghia), pp. 1–13, ASME.
Stuart, J. T. & Diprima, R. C. 1978 The Eckhaus and Benjamin-Feir resonance mechanisms. Proc. R. Soc. Lond. A 362, 27.Google Scholar
Takamoto, M. 1987 A study of the wake structure behind bluff rings. Bull. NRLM (Japan) 36, 441.Google Scholar
Takamoto, M. & Izumi, K. 1981 Experimental observation of stable arrangement of vortex rings. Phys. Fluids 24, 1582.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.Google Scholar
Tuckerman, L. & Barkley, D. 1990 Bifurcation analysis of the Eckhaus instability. Physica D 46, 57.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
Williamson, C. H. K. 1988a Defining a universal and continuous Strouhal–Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids 31, 2742.Google Scholar
Williamson, C. H. K. 1988b The existence of two stages in the transition to three-dimensionality of a circular cylinder wake. Phys. Fluids 31, 3165.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. 1991 Three-dimensional aspects and transition of the wake of a circular cylinder. In Turbulent Shear Flows 7 (ed. F. Durst & J. Launder), pp. 173–194. Springer.
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. 1995 Vortex dynamics in the wake of a cylinder. In Fluid Vortices (ed. S. Green). Kluwer Academic (to appear).
Williamson, C. H. K. & Roshko, A. 1990 Measurements of base pressure in the wake of a cylinder at low Reynolds numbers. Z. Flugwiss. Weltraumforsch. 14, 38.Google Scholar
Zhang, H.-Q., Noack, B. R. & Eckelmann, H. 1994 Numerical computation of the 3-D cylinder wake. Rep. 3/1994. Max-Planck-Institut für Strömungforschung, Göttingen.