Skip to main content

On the origin of the flip–flop instability of two side-by-side cylinder wakes

  • M. Carini (a1), F. Giannetti (a2) and F. Auteri (a1)

In this work the flip–flop instability occurring in the flow past two side-by-side circular cylinders is numerically investigated within the range of non-dimensional gap spacing $0.6 and Reynolds number $50 . The inherent two-dimensional flow pattern is characterized by an asymmetric unsteady wake (with respect to the horizontal axis of symmetry) with the gap flow being deflected alternatively toward one of the cylinders. Such behaviour has been ascribed by other authors to a bistability of the flow, and therefore termed flip–flop. In contrast, the simulations performed herein provide new evidence that at low Reynolds numbers the flip–flopping state develops through an instability of the in-phase synchronized vortex shedding between the two cylinder wakes. This new scenario is confirmed and explained by means of a linear global stability investigation of the in-phase periodic base flow. The Floquet analysis reveals indeed that a pair of complex-conjugate multipliers becomes unstable having the same low frequency as the gap flow flip-over. The neutral curve of this secondary instability is tracked within the above range of gap spacing. The spatiotemporal shape of the unstable Floquet mode is then analysed and its structural sensitivity is considered in order to identify the ‘core’ region of the flip–flop instability mechanism.

Corresponding author
Email address for correspondence:
Hide All
Åkervik, E., Brandt, L., Henningson, D. S., Hoepffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102.
Afgan, I., Kahil, Y., Benhamadouche, S. & Sagaut, P. 2011 Large eddy simulation of the flow around single and two side-by-side cylinders at subcritical Reynolds numbers. Phys. Fluids 23, 075101.
Akinaga, T. & Mizushima, J. 2005 Linear stability of flows past two circular cylinders in a side-by-side arrangement. J. Phys. Soc. Japan 74 (5), 13661369.
Bearman, P. W. & Wadcock, A. J. 1973 The interaction between a pair of circular cylinders normal to a stream. J. Fluid Mech. 61, 499511.
Bittanti, S. & Colaneri, P. 2009 Periodic Systems: Filtering and Control (Communication and Control Engineering). Springer.
Camarri, S. & Giannetti, F. 2010 Effect of confinement on three-dimensional stability in the wake of a circular cylinder. J. Fluid Mech. 642, 477487.
Camarri, S. & Iollo, A. 2010 Feedback control of the vortex-shedding instability based on sensitivity analysis. Phys. Fluids 22, 094102.
Chen, L., Tu, J. Y. & Yeoh, G. H. 2003 Numerical simulation of turbulent wake flows behind two side-by-side cylinders. J. Fluids Struct. 18, 387403.
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 156, 209240.
Coddington, E. & Levinson, N. 1955 Theory of Ordinary Differential Equations. McGraw-Hill.
Davis, T. A. 2004 Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30 (2), 196199.
Drazin, P. G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.
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.
Giannetti, F., Fabre, D., Tchoufag, J. & Luchini, P. 2012 The steady oblique path of buoyancy-driven rotating spheres. In 9th European Fluid Mechanics Conference, Rome, 9–13 September 2012 .
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid. Mech. 581, 167197.
Giannetti, F., Luchini, P. & Marino, L. 2009 Characterization of the three-dimensional instability in a lid-driven cavity by an adjoint-based analysis. In Seventh IUTAM Symposium on Laminar–Turbulent Transition (ed. Schlatter, P. & Henningson, D. S.), pp. 9697. KTH.
Haque, S., Lashgari, I., Brandt, L. & Giannetti, F. 2012 Stability of fluids with shear-dependent viscosity in the lid-driven cavity. J. Non-Newtonian Fluid Mech. 173, 4961.
Ilak, M., Schlatter, P., Bagheri, S. & Henningson, D. 2011 Bifurcation and stability analysis of a jet in crossflow. Part 1: Onset of global instability at a low velocity ratio. J. Fluid Mech. 696, 94121.
Kang, S. 2003 Characteristics of flow over two circular cylinders in a side-by-side arrangement at low Reynolds numbers. Phys. Fluids 15, 24862498.
Kim, H. J. & Durbin, P. A. 1988 Investigation of the flow between a pair of circular cylinders in the flopping regime. J. Fluid Mech. 196, 431448.
Lashgari, I., Pralits, J. O., Giannetti, F. & Brandt, L. 2012 First instability of the flow of shear-thinning and shear-thickening fluids past a circular cylinder. J. Fluid Mech. 701, 201227.
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users Guide. SIAM.
Luchini, P. 2011 Private communication.
Luchini, P., Giannetti, F. & Pralits, J. O. 2008 Structural sensitivity of linear and nonlinear global modes. In Proceedings of the 5th AIAA Theoretical Fluid Mechanics Conference, 23–26 June, Seattle, Washington pp. 119. Curran Associates Inc.
Luchini, P., Pralits, J. O. & Giannetti, F. 2007 Structural sensitivity of the finite-amplitude vortex shedding behind a circular cylinder. In Proceedings of the 2nd IUTAM Symposium on unsteady separated flows and their control, 18–22 June 2007, Corfu, Greece (ed. Braza, M. & Houringan, K.), pp. 151160. Springer.
Lust, K., Roose, D., Spence, A. & Champneys, A. R. 1998 An adaptive Newton–Picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comp. 19 (4), 11181209.
Marino, L. & Luchini, P. 2009 Adjoint analysis of the flow over a forward-facing step. Theor. Comp. Fluid Dyn. 23, 3754.
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.
Meliga, P., Sipp, D. & Chomaz, J.-M. 2010 Effect of compressibility on the global stability of axisymmetric wake flows. J. Fluid Mech. 660, 499526.
Mizushima, J. & Ino, Y. 2008 Stability of flows past a pair of circular cylinders in a side-by-side arrangement. J. Fluid Mech. 595, 491507.
Peschard, I. & Le Gal, P. 1996 Coupled wakes of cylinders. Phys. Rev. Lett. 77, 31223125.
Pralits, J. O., Brandt, L. & Giannetti, F. 2010 Instability and sensitivity of the flow around a rotating circular cylinder. J. Fluid Mech. 650, 124.
Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comp. Phys. 96, 1553.
Robichaux, J., Balanchadar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the wake of a square cylinder. Phys. Fluids 11, 560578.
Shroff, G. M. & Keller, H. B. 1993 Stabilization of unstable procedures: the recursive projection method. SIAM J. Numer. Anal. 30 (4), 10991120.
Sumner, D. 2010 Two circular cylinders in cross-flows: a review. J. Fluids Struct. 26, 849899.
Sumner, D., Wong, S. S. T., Price, S. J. & Païdoussis, M. P. 1999 Fluid behavior of side-by-side circular cylinders in steady cross-flow. J. Fluids Struct. 13, 309338.
Trottenberg, U., Oosterlee, C. & Schüller, A. 2001 Multigrid. Academic Press.
Viaud, B., Serre, E. & Chomaz, J.-M. 2011 Transition to turbulence through steep global-modes cascade in an open rotating cavity. J. Fluid Mech. 688, 493506.
Wang, Z. J., Zhou, Y. & Li, H. 2002 Flow-visualization of a two side-by-side cylinder wake. J. Flow Visual. Image Process. 9, 123138.
Williamson, C. H. K. 1985 Evolution of a single wake behind a pair of bluff bodies. J. Fluid Mech. 159, 118.
Zdravkovich, M. M. 1977 Review of flow interference between two circular cylinders in various arrangement. Trans. ASME I: J. Fluids Engng 99, 618633.
Zhou, Y., Zhang, H. J. & Yiu, M. W. 2002 The turbulent wake of two side-by-side circular cylinders. J. Fluid Mech. 458, 303332.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 121 *
Loading metrics...

Abstract views

Total abstract views: 278 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th March 2018. This data will be updated every 24 hours.