Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-09T21:52:51.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-periodic cylinder wakes and the Ginzburg–Landau model

Published online by Cambridge University Press:  26 April 2006

Pierre Albarède  and
Michel Provansal
Pierre Albarède
Affiliation:
Laboratoire de Recherche en Combustion, Université de Provence-CNRS, 13397 Marseille CEDEX 13, France Present address: Centre d'Etude de Cadarache 13108 St Paul les Durance, France.
Michel Provansal
Affiliation:
Laboratoire de Recherche en Combustion, Université de Provence-CNRS, 13397 Marseille CEDEX 13, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The time-periodic phenomena occurring at low Reynolds numbers (Re [lsim ] 180) in the wake of a circular cylinder (finite-length section) are well modelled by a Ginzburg–Landau (GL) equation with zero boundary conditions (Albarède & Monkewitz 1992). According to the GL model, the wake is mainly governed by a rescaled length, based on the aspect ratio and the Reynolds number. However, the determination of coefficients is not complete: we correct a former evaluation of the nonlinear Landau coefficient, we show difficulties in obtaining a consistent set of coefficients for different Reynolds numbers or end configurations, and we propose the use of an ‘influential’ length. New two-point velocimetry results are presented: phase measurements show that a subtle property is shared by the three-dimensional wake and the GL model.

Two time-quasi-periodic phenomena – the second mode observed for smaller aspect ratios, and the dislocated chevron observed for larger aspect ratios – are presented and precisely related to the GL model. Only the linear characteristics of the second mode are readily explained; its existence depends on the end conditions. Moreover, through a quasi-static variation of the length, the second mode evolves continuously to end cells (and vice versa). Observations of the dislocated chevron are recalled. A very similar instability is found on the chevron solution of the GL equation, when the model parameters (c1, c2) move towards the phase diffusion unstable region. The early stages of this instability are qualitatively similar to the observed patterns.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 291 , 25 May 1995 , pp. 191 - 222
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. 1989 Perturbations tridimensionnelles du sillage d'un cylindre à bas nombre de Reynolds. Stage de Magistère Interuniversitaire de Physique, Université Paris 6.
Albarède, P. 1991 Auto-organisation dans le sillage 3D d'un obstacle non profilé. Thèse Université de Provence. (English version available from LRC: Self-organization in the 3D wakes of bluff bodies.)
Albarède, P., Leweke, T. & Provansal, M. 1992 The Ginzburg-Landau equation as a model of the three-dimensional circular cylinder at low Reynolds numbers. IUTAM Symp. on Bluff Body Wakes, Dynamics and Instabilities, September 1992, Göttingen, Germany, pp. 7–11.
Albarède, P. & Monkewitz, P. A. 1992 A model for the formation of oblique shedding and ‘chevron’ patterns in cylinder wakes. Phys. Fluids A 4, 744–756.Google Scholar
Albarède, P., Provansal, M. & Boyer, L. 1990 Modélisation par l'équation de Ginzburg-Landau du sillage tridimensionnel d'un obstacle allongé. C. R. Acad. Sci. Paris 310 (II), 459–464.Google Scholar
Berger, E. 1967 Suppression of vortex shedding and turbulence behind oscillating cylinders. Phys. Fluids suppl. S191–S193.
Berger, E. & Wille, R. 1972 Periodic flow phenomena. Ann. Rev. Fluid Mech. 4, 313–340.Google Scholar
Chiffaudel, A. 1992 Non-linear stability analysis of two-dimensional patterns in the wake of a circular cylinder. Europhys. Lett. 18, 589–594.Google Scholar
Davey, A., Hocking, L. M. & Stewartson, K. 1974 On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 63, 529–536.Google Scholar
Dusěk, J., Gal, P. Le & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.Google Scholar
Gaster, M. 1969 Vortex shedding from slender cones at low Reynolds number. J. Fluid Mech. 38, 565–576.Google Scholar
Gaster, M. 1971 Vortex shedding from circular cylinders at low Reynolds number. J. Fluid Mech. 46, 749–756.Google Scholar
Gerich, D. 1987 A limiting process for the von Kármaán vortex street showing the change from two-to three-dimensional flow. Proc. 4th Intl Symp. on Flow Visualization.
Gerich, D. & Eckelmann, H. 1982 Influence of end plates and free ends on the shedding frequency of circular cylinders. J. Fluid Mech. 122, 109–121.Google Scholar
Gerrard, J. H. 1966 The three-dimensional structure of the wake of a circular cylinder. J. Fluid Mech. 25, 143–164.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
Hama, F. R. 1957 Three-dimensional vortex pattern behind a circular cylinder. J. Aeronaut. Sci. 24, 156.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537.Google Scholar
Jenffer, P., Wesfreid, J. E., Goujon-Durand, S., Bénard, R. & Boussora, R. 1992 Spatial properties along the flow of the vortex emission behind a bluff body. Proc. IUTAM Symp. on Bluff-Body Wakes, Dynamics, and Instabilities, Göttingen, Sept. 7–11.
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, 1607–1614.Google Scholar
Kuramoto, Y. 1984 Chemical Oscillations, Waves, and Turbulence. Springer.
Landau, L. & Lifchitz, E. 1971 Mécanique des fluides. Moscow: Mir.
Dizes, S. Le, Monkewitz, P. A. & Huerre, P. 1992 Weakly nonlinear analysis of spatially developing shear flows. Proc. IUTAM Symp. on Bluff-Body Wakes, Dynamics, and Instabilities, Göttingen, Sept. 7–11.
Lee, T. & Budwig, R. 1991 A study of the effect of aspect ratio on vortex shedding behind circular cylinders. Phys. Fluids A 3, 309–315.Google Scholar
Leweke, T. & Provansal, M. 1994 Determination of the parameters of the Ginzburg-Landau wake model from experiments on a bluff ring. Europhys. Lett. 27, 655–660.Google Scholar
Leweke, T., Provansal, M. & Boyer, L. 1993 Sillage tridimensionnel d'un obstacle torique et modélisation par l'equation de Ginzburg-Landau. C.R. Acad. Sci. Paris 316 (II), 287–292.Google Scholar
Mathis, C. 1983 Propriétés des composantes de vitesse transverses dans l’écoulement de Bénard – von Kármán aux faibles nombres de Reynolds. Thèse, Université de Provence.
Mathis, C., Provansal, M. & Boyer, L. 1984a The Bénard-von Kármán instability: an experimental study near the threshold. J. Phys. Lett. Paris 45 L-483–L-491.Google Scholar
Mathis, C., Provansal, M. & Boyer, L. 1984b The rotating grating applied to the study of the Bénard-von Kármán instability near the threshold. Proc. Second Intl Symp. on Applications of Laser Anemometry to Fluid Mechanics, Lisbon.
Monkewitz, P. A. 1988 The absolute and convective nature of instability in two-dimensional wakes at low Reynolds number. Phys. Fluids 31, 999–1006.Google Scholar
Noack, B. & Eckelmann, H. 1991 Two-dimensional, viscous, incompressible flow around a circular cylinder. Max Planck Institut für Strömungsforschung, Göttingen, Bericht 104/1991.
Noack, B., Ohle, F. & Eckelmann, H. 1991 On cell formation in vortex streets. J. Fluid Mech. 227, 293–308.Google Scholar
Nozaki, B. & Bekki, N. 1984 Exact solutions of the generalized Ginburg-Landau equation. J. Phys. Soc. Japan 53, 1581–1582.Google Scholar
Papangelou, A. 1992 Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 242, 299–321.Google Scholar
Park, D. S. & Redekopp, L. G. 1992 Selection principles for spatio-temporal patterns in wake flows. Proc. IUTAM Symp. on Bluff-Body Wakes, Dynamics, and Instabilities, Göttingen, Sept. 7–11.
Provansal, M. 1988 Etude expérimentale de l'instabilité de Bénard-von Kármán. Thèse, Université de Provence.
Provansal, M., Leweke, T. & Albaréde, P. 1992 Visualisation du sillage d'un obstacle au moyen de nappes de fumée: mise en évidence des effets tridimensionnels. 5éme Colloque National de Visualisation et de Traitement d'Images en Mécanique des Fluides, 2–5 Juin 1992, Poitiers.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.Google Scholar
Rossi, M., Huerre, P. & Redekopp, L. G. 1989 A model of boundary effects in Kármán vortex streets. Bull. Am. Phys. Soc. 34, 2282.Google Scholar
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of twodimensional bluff bodies and their control. J. Fluid Mech. 271, 17–53.Google Scholar
Shraiman, B. I., Pumir, A., Saarloos, W. Hohenberg, P. C. Van & Chaté, H. 1992 Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. Physica ([A-Z]) 57, 241–248.Google Scholar
Slaouti, A. & Gerrard, J. H. 1981 An experimental investigation of the end effects on the wake of a circular cylinder towed through water at low Reynolds numbers. J. Fluid Mech. 112, 297–314.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. 41–67. Springer.
Strykowski, P. J. 1986 The control of absolutely and convectively unstable flows. PhD thesis, Yale University.
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6, 547–567.Google Scholar
Tritton, D. J. 1971 A note on vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 45, 203–208.Google Scholar
Atta, C. W. Van & Gharib, M. 1987 Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers. J. Fluid Mech. 174, 113–133.Google Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31, 3165–3168.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, 579–627.Google Scholar