Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T14:25:41.757Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional Floquet stability analysis of the wake of a circular cylinder

Published online by Cambridge University Press:  26 April 2006

Dwight Barkley  and
Ronald D. Henderson
Dwight Barkley
Affiliation:
Nonlinear Systems Laboratory, Mathematics Institute. University of Warwick, Coventry, CV4 7AL, UK
Ronald D. Henderson
Affiliation:
Aeronautics and Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 322 , 10 September 1996 , pp. 215 - 241
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

Amon, C. H. & Patera, A. T. 1989 Numerical calculation of stable three-dimensional tertiary states in grooved-channel flow.. Phys. Fluids A 1, 2005–2009.Google Scholar
Barkley, D. 1990 Theory and predictions for finite-amplitude waves in 2-dimensional plane Poiseuille flow.. Phys. Fluids A 2, 955–970.Google Scholar
Goldhirsch, I., Orszag, S. A. & Maulik, B. K. 1987 An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices. J. Sci. Comput. 2, 33–58.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hammache, M. & Gharib, M. 1991 An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mech. 232, 567–590.Google Scholar
Henderson, R. D. 1994 Unstructured Spectral Element Methods: Parallel Algorithms and Simulations. PhD thesis Princeton University.
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8, 1683–1685.Google Scholar
Henderson, R. D. & Karniakes, G. E. 1995 Unstructured spectral element methods for simulation of turbulent flows. J. Comput. Phys. 122, 191–217.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
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–413.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1989 Frequency selection and asymptotic states in laminar wakes. J. Fluid Mech. 199, 441–469.Google Scholar
Karniadakis, G. E. & Triantafyllou, G. S. 1992 Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech. 238, 1–30.Google Scholar
Lasheras, J. C. & Meiburg, E. 1990 Three-dimensional vorticity modes in the wake of a flat plate.. Phys. Fluids A 2, 371–380.Google Scholar
Leweke, T. & Provansal, M. 1994 Model for the transition in bluff-body wakes. Phys. Rev. Lett. 72, 3174–3177.Google Scholar
Leweke, T. & Provansal, M. 1995 The flow behind rings - bluff-body wakes without end effects. J. Fluid Mech. 288, 265–310.Google Scholar
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.Google Scholar
Mansy, H., Yang, P.-M. & Williams, D. R. 1994 Quantitative measurements of three-dimensional structures in the wake of a circular cylinder. J. Fluid Mech. 270, 277–296.Google Scholar
Mathis, C., Provansal, M. & Boyer, L. 1987 Bènard-von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.Google Scholar
Meiburg, E. & Lasheras, J. C. 1988 Experimental and numerical investigation of the three-dimensional transition in plane wakes. J. Fluid Mech. 190, 1–37.Google Scholar
Miller, G. D. & Williamson, C. H. K. 1994 Control of three-dimensional phase dynamics in a cylinder wake. Exps. Fluids 18, 26–35.Google Scholar
Mittal, R. & Balachandar, S. 1995 Generation of streamwise vortical structures in bluff body wakes. Phys. Rev. Lett. 75, 1300–1303.Google Scholar
Noack, B. R. & Eckelmann, H. 1994a A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.Google Scholar
Noack, B. R. & Eckelmann, H. 1994b A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder.. Phys. Fluids A 6, 124–143.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, 1279–1281.Google Scholar
Norberg, C. 1994 An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287–316.Google Scholar
Roshko, A. 1993 Perspectives on bluff body aerodynamics. J. Wind Engng 49, 79–100.Google Scholar
Schatz, M. F., Barkley, D. & Swinney, H. L. 1995 Instability in spatially periodic open flow. Phys. Fluids 7, 344–358.Google Scholar
Thompson, M., Hourigan, K. & sheridan, J. 1994 Three-dimensional instabilities in the cylinder wake. In Intl Colloq. Jets, Wakes, and Shear Layers. Melbourne, Australia. CSIRO.
Thompson, M., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12, 190–196.Google Scholar
Tomboulides, A. G., Triantafyllou, G. S. & Karniadakis, G. E. 1992 A new mechanism of period doubling in free shear flows.. Phys. Fluids A 4, 1329–1332.Google Scholar
Watkins, D. S. 1993 Some perspectives on the eigenvalue problem. SIAM Rev. 35, 430–471.Google Scholar
Williams, D. R., Mansy, H. & Abouel-Fotouh, A. 1996 Three-dimensional subharmonic waves during transition in the near-wake region of a cylinder. Phys. Fluids 8, 1476–1485.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
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, 393–441.Google Scholar
Williamson, C. H. K. 1996a “Mode A” secondary instability in wake transition. Phys. Fluids 8, 1680–1682.Google Scholar
Williamson, C. H. K. 1996b Vortex dynamics in the cylinder wake. Ann. Rev. Fluid Mech. 28, 477–539.Google Scholar
Williamson, C. H. K. 1996c Three-dimensional transition in the cylinder wake. J. Fluid Mech. (to appear).Google Scholar
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, 38–46.Google Scholar
Wu, J., Sheridan, J., Soria, J. & Welsh, M. C. 1994 An experimental investigation of streamwise vortices in the wake of a bluff body. J. Fluid Struct. 8, 621–625.Google Scholar
Wu, J., Sheridan, J., Welsh, M. C. & Hourigan, K. 1996 Three-dimensional vortex structures in a cylinder wake. J. Fluid Mech. 312, 201–222Google Scholar
Zhang, H.-Q., Fey, U., Noack, B. R., KÖnig, M. & Eckelmann, H. 1995 On the transition of the cylinder wake. Phys. Fluids 7, 779–794.Google Scholar