Skip to main content Accessibility help
×
Home

Stability analysis of the elliptic cylinder wake

  • Justin S. Leontini (a1), David Lo Jacono (a2) (a3) (a4) and Mark C. Thompson (a2)

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.

Copyright

Corresponding author

Email address for correspondence: justin.leontini@gmail.com

References

Hide All
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.
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), 52475252.
Blackburn, H. M., Marques, F. & Lopez, J. M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395411.
Canuto, C., Hussaini, M., Quarteroni, A. & Zang, T. 1990 Spectral Methods in Fluid Dynamics, 2nd edn. Springer.
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, 5980.
Giannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319337.
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2009 Pulsatile flow in stenotic geometries: flow behaviour and stability. J. Fluid Mech. 622, 291320.
Henderson, R. D. 1997 Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech. 352, 65112.
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 2345.
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Karniadakis, G. Em., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods of the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414443.
Karniadakis, G. Em. & Sherwin, S. J. 2005 Spectral/HP Methods for Computational Fluid Dynamics. Oxford University Press.
Karniadakis, G. E. & Triantafyllou, G. S. 1992 Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech. 238, 130.
Kuo, Y.-H. & Baldwin, L. V. 1967 The formation of elliptical wakes. J. Fluid Mech. 27, 353360.
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30 (8), 23392342.
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, 445457.
Leontini, J. S., Thompson, M. C. & Hourigan, K. 2007 Three-dimensional transition in the wake of a transversely oscillating cylinder. J. Fluid Mech. 577, 79104.
Leweke, T. & Williamson, C. H. K. 1998 Three-dimensional instabilities in wake transition. Eur. J. Mech. (B/Fluids) 17 (4), 571586.
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, 349362.
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf-bifurcation in spherical Couette flow. Phys. Fluids 7, 8091.
Mittal, R. & Balachandar, S. 1995 Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders. Phys. Fluids 7 (8), 18411865.
Modi, V. J. & Dikshit, A. K. 1975 Near-wakes of elliptic cylinders in subcritical flow. AIAA J. 13 (4), 490497.
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 122.
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, 379391.
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, 129.
Robichaux, J., Balachandar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the wake of a square cylinder. Phys. Fluids 11 (3), 560578.
Roshko, A. 1955 On the wake and drag of bluff bodies. J. Aero. Sci. 22, 124132.
Roshko, A. 1993 Perspectives on bluff body aerodynamics. J. Wind Engng Ind. Aerodyn. 49, 79100.
Ryan, K., Thompson, M. C. & Hourigan, K. 2005 Three-dimensional transition in the wake of bluff elongated cylinders. J. Fluid Mech. 538, 129.
Sheard, G. J., Fitzgerald, M. J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 4369.
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), L68L71.
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, 333358.
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic instability. J. Fluid Mech. 4, 121.
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Japan 11, 302307.
Thompson, M. C., Hourigan, K., Cheung, A. & Leweke, T. 2006 Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 30 (11), 13561369.
Thompson, M. C., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12, 190196.
Thompson, M. C. & Le Gal, P. 2004 The Stuart–Landau model applied to wake transition revisited. Eur. J. Mech. (B/Fluids) 23 (1), 219228.
Thompson, M. C., Leweke, T. & Hourigan, K. 2007 Sphere–wall collisions: vortex dynamics and stability. J. Fluid Mech. 575, 121148.
Thompson, M. C., Leweke, T. & Williamson, C. H. K. 2001 The physical mechanism of transition in bluff body wakes. J. Fluid Struct. 15, 607616.
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, 570600.
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. 453466. Springer.
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31 (11), 31653168.
Williamson, C. H. K. 1996 Three-dimensional wake transition. J. Fluid Mech. 328, 345407.
Zdravkovich, M. M. 1997 Flow Around Circular Cylinders: Volume 1: Fundamentals. Oxford University Press.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed