Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-09T21:40:18.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation for flow past a cylinder between parallel planes

Published online by Cambridge University Press:  26 April 2006

J.-H. Chen ,
W. G. Pritchard  and
S. J. Tavener
J.-H. Chen
Affiliation:
Department of Naval Architecture, National Taiwan Ocean University, 2 Pei-Ning Rd, Keelung 202, Taiwan, ROC
W. G. Pritchard
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, USA
S. J. Tavener
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.

A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.

A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.

Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 23 - 41
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

Amick, C. J. 1977 Steady solutions of the Navier–Stokes equations in unbounded channels and pipes. Ann. Scuola Norm. Sup. Pisa IV 4, 473–513.Google Scholar
Amick, C. J. 1978 Properties of steady Navier–Stokes solutions for certain unbounded channels and pipes. Nonlinear Anal. 2, 689–720.Google Scholar
Amick, C. J. & Fraenkel, L. E. 1980 Steady solutions of the Navier–Stokes equations representing plane flow in channels of various types. Acta Mathematica 144, 83–152.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bercovier, M. & Engelman, M. 1979 A finite element method for the numerical solution of viscous incompressible flows. J. Comput. Phys. 30, 181–201.Google Scholar
Chen, J.-H. 1990 A numerical study of flow past a cylindrical body. PhD dissertation, Penn State University.
Cliffe, K. A., Garratt, T. J. & Spence, A. 1993 A modified Cayley transform for the discretized Navier–Stokes equations. In Proc. ISNA '92, Applications of Mathematics, vol. 38, pp. 281–288.
Coutanceau, M. & Bouard, R. 1977a Experimental determination of the main features of viscous flow, in the wake of a circular cylinder in uniform translation. Part 1. Steady flow. J. Fluid Mech. 79, 231–256.Google Scholar
Coutanceau, M. & Bouard, R. 1976b Experimental determination of the main features of viscous flow in the wake of a circular cylinder in uniform translation. Part 2. Unsteady flow. J. Fluid Mech. 79, 257–272.Google Scholar
Crouzeix, M. & Raviart, P.-A. 1973 Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO R-3, 33–76.Google Scholar
Engelman, M. 1982 FIDAP – A fluid dynamics analysis package. Adv. Engng Software 4, 163–166.Google Scholar
Fearn, R. M., Mullin, T. & Cliffe, K. A. 1990 Nonlinear flow phenomena in a sudden symmetric expansion. J. Fluid Mech. 211, 595–608.Google Scholar
Gerrard, J. H. 1978 The wakes of cylindrical bluff bodies at low Reynolds number. Phil. Trans. R. Soc. Lond. A 288, 351–382.Google Scholar
Griewank, A. & Reddien, G. 1983 The calculation of Hopf points by a direct method. IMA J. Numer. Anal. 3, 295–304.Google Scholar
Gunzburger, M. D. 1989 Finite Element Methods for Viscous Incompressible Flows. Academic.
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
Kim, I. & Pearlstein, A. J. 1990 Stability of flow past a sphere. J. Fluid Mech. 211, 73–93.Google Scholar
Lodge, A. S., Pritchard, W. G. & Scott, L. R. 1991 The hole-pressure problem. IMA J. Appl. Maths. 46, 39–66.Google Scholar
Nakamura, I. 1976 Steady wake behind a sphere. Phys. Fluids 19, 5–8.Google Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.Google Scholar
Nishioka, M. & Sato, H. 1974 Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 65, 97–112.Google Scholar
Pritchard, W. G. 1983 Some viscous-dominated flows. In Trends and Applications of Pure Mathematics to Mechanics (ed. P. G. Ciarlet & M. Rousseau), pp. 305–332. Springer.
Pritchard, W. G., Scott, L. R. & Tavener, S. J. 1992 Numerical and asymptotic methods for certain viscous free-surface flows. Phil. Trans. R. Soc. Lond. A 340, 1–45.Google Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kàrmàn instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.Google Scholar
Taneda, S. 1956a Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Japan 11, 302–307.Google Scholar
Taneda, S. 1956b Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Japan 11, 1104–1108.Google Scholar
Tavener, S. J. 1994 Bifurcation of the O(2)-symmetric flow past a sphere in a pipe. To appear in Phys. Fluids.
Taylor, C. & Hood, P. 1973 A numerical solution of the Navier–Stokes equations using the finite element technique. Comput. Fluids 1, 73–100.Google Scholar
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 6, 547–567.Google Scholar
Winters, K. H. 1985 ENTWIFE user manual (release 1). Harwell Rep. AERE-R 11577.
Wu, J. S. & Faeth, G. M. 1993 Sphere wakes in still surroundings at intermediate Reynolds numbers. AIAA J. 31, 1448–1455.Google Scholar