Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T04:54:10.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability analysis of the elliptic cylinder wake

Published online by Cambridge University Press:  16 December 2014

Justin S. Leontini ,
David Lo Jacono  and
Mark C. Thompson
Justin S. Leontini*
Affiliation:
Department of Mechanical and Product Design Engineering, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
David Lo Jacono
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia Université de Toulouse; INP; IMFT (Institut de Mécanique des Fluides de Toulouse), Allée Camille Soula, F-31400 Toulouse, France CNRS; IMFT; F-31400 Toulouse, France
Mark C. Thompson
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia
*
†Email address for correspondence: justin.leontini@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents the results of numerical stability analysis of the wake of an elliptical cylinder. Aspect ratios where the ellipse is longer in the streamwise direction than in the transverse direction are considered. The focus is on the dependence on the aspect ratio of the ellipse of the various bifurcations to three-dimensional flow from the two-dimensional Kármán vortex street. It is shown that the three modes present in the wake of a circular cylinder (modes A, B and QP) are present in the ellipse wake, and that in general they are all stabilized by increasing the aspect ratio of the ellipse. Two new pertinent modes are found: one long-wavelength mode with similarities to mode A, and a second that is only unstable for aspect ratios greater than approximately 1.75, which has similar spatiotemporal symmetries to mode B but has a distinct spatial structure. Results from fully three-dimensional simulations are also presented confirming the existence and growth of these two new modes in the saturated wakes.

JFM classification

Instability: Instability Wakes/Jets: Vortex streets Wakes/Jets: Wakes/Jets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 763 , 25 January 2015 , pp. 302 - 321
Check for updates
Copyright
© 2014 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

Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Barkley, D., Tuckerman, L. S. & Golubitsky, M. 2000 Bifurcation theory for three-dimensional flow in the wake of a circular cylinder. Phys. Rev. E 61 (5), 5247–5252.Google Scholar
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411.Google Scholar
Canuto, C., Hussaini, M., Quarteroni, A. & Zang, T. 1990 Spectral Methods in Fluid Dynamics, 2nd edn. Springer.Google Scholar
Dušek, J., Le Gal, P. & 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
Giannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319–337.CrossRefGoogle Scholar
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2009 Pulsatile flow in stenotic geometries: flow behaviour and stability. J. Fluid Mech. 622, 291–320.CrossRefGoogle Scholar
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352, 65–112.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
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Karniadakis, G. Em., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods of the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443.Google Scholar
Karniadakis, G. Em. & Sherwin, S. J. 2005 Spectral/HP Methods for Computational Fluid Dynamics. Oxford University Press.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
Kuo, Y.-H. & Baldwin, L. V. 1967 The formation of elliptical wakes. J. Fluid Mech. 27, 353–360.Google Scholar
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30 (8), 2339–2342.Google Scholar
Le Gal, P., Nadim, A. & Thompson, M. 2001 Hysteresis in the forced Stuart–Landau equation: application to vortex shedding from an oscillating cylinder. J. Fluids Struct. 15, 445–457.Google Scholar
Leontini, J. S., Thompson, M. C. & Hourigan, K. 2007 Three-dimensional transition in the wake of a transversely oscillating cylinder. J. Fluid Mech. 577, 79–104.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Three-dimensional instabilities in wake transition. Eur. J. Mech. (B/Fluids) 17 (4), 571–586.Google Scholar
Lo Jacono, D., Leontini, J. S., Thompson, M. C. & Sheridan, J. 2010 Modification of three-dimensional transition in the wake of a rotationally oscillating cylinder. J. Fluid Mech. 643, 349–362.Google Scholar
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf-bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.Google Scholar
Mittal, R. & Balachandar, S. 1995 Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders. Phys. Fluids 7 (8), 1841–1865.CrossRefGoogle Scholar
Modi, V. J. & Dikshit, A. K. 1975 Near-wakes of elliptic cylinders in subcritical flow. AIAA J. 13 (4), 490–497.Google Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.Google Scholar
Rao, A., Leontini, J., Thompson, M. C. & Hourigan, K. 2013a Three-dimensionality in the wake of a rapidly rotating cylinder in a uniform flow. J. Fluid Mech. 730, 379–391.Google Scholar
Rao, A., Leontini, J., Thompson, M. C. & Hourigan, K. 2013b Three-dimensionality in the wake of a rotating cylinder in a uniform flow. J. Fluid Mech. 717, 1–29.Google Scholar
Robichaux, J., Balachandar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the wake of a square cylinder. Phys. Fluids 11 (3), 560–578.Google Scholar
Roshko, A. 1955 On the wake and drag of bluff bodies. J. Aero. Sci. 22, 124–132.CrossRefGoogle Scholar
Roshko, A. 1993 Perspectives on bluff body aerodynamics. J. Wind Engng Ind. Aerodyn. 49, 79–100.Google Scholar
Ryan, K., Thompson, M. C. & Hourigan, K. 2005 Three-dimensional transition in the wake of bluff elongated cylinders. J. Fluid Mech. 538, 1–29.Google Scholar
Sheard, G. J., Fitzgerald, M. J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 43–69.Google Scholar
Sheard, G. J., Thompson, M. C. & Hourigan, K. 2003 A coupled Landau model describing the Strouhal–Reynolds number profile of the three-dimensional wake of a circular cylinder. Phys. Fluids Lett. 15 (9), L68–L71.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic instability. J. Fluid Mech. 4, 1–21.Google Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Japan 11, 302–307.Google Scholar
Thompson, M. C., Hourigan, K., Cheung, A. & Leweke, T. 2006 Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 30 (11), 1356–1369.Google Scholar
Thompson, M. C., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12, 190–196.Google Scholar
Thompson, M. C. & Le Gal, P. 2004 The Stuart–Landau model applied to wake transition revisited. Eur. J. Mech. (B/Fluids) 23 (1), 219–228.Google Scholar
Thompson, M. C., Leweke, T. & Hourigan, K. 2007 Sphere–wall collisions: vortex dynamics and stability. J. Fluid Mech. 575, 121–148.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Williamson, C. H. K. 2001 The physical mechanism of transition in bluff body wakes. J. Fluid Struct. 15, 607–616.Google Scholar
Thompson, M. C., Radi, A., Rao, A., Sheridan, J. & Hourigan, K. 2014 Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate. J. Fluid Mech. 751, 570–600.Google Scholar
Tuckerman, L. S. & Barkley, D. 2000 Numerical methods for bifurcation problems and large-scale dynamical systems. In Bifurcation Analysis for Timesteppers, IMA Volumes in Mathematics and its Application, vol. 119, pp. 453–466. Springer.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 (11), 3165–3168.Google Scholar
Williamson, C. H. K. 1996 Three-dimensional wake transition. J. Fluid Mech. 328, 345–407.CrossRefGoogle Scholar
Zdravkovich, M. M. 1997 Flow Around Circular Cylinders: Volume 1: Fundamentals. Oxford University Press.Google Scholar