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Centre-manifold reduction of bifurcating flows

Published online by Cambridge University Press:  12 February 2015

M. Carini ,
F. Auteri  and
F. Giannetti
M. Carini
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
F. Auteri*
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
F. Giannetti
Affiliation:
Dipartimento di Ingegneria Industriale, Università degli studi di Salerno, via Ponte don Melillo, 84084 Fisciano (SA), Italy
*
†Email address for correspondence: franco.auteri@polimi.it
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Abstract

In this paper we describe a general and systematic approach to the centre-manifold reduction and normal form computation of flows undergoing complicated bifurcations. The proposed algorithm is based on the theoretical work of Coullet & Spiegel (SIAM J. Appl. Maths, vol. 43(4), 1983, pp. 776–821) and can be used to approximate centre manifolds of arbitrary dimension for large-scale dynamical systems depending on a scalar parameter. Compared with the classical multiple-scale technique frequently employed in hydrodynamic stability, the proposed method can be coded in a rather general way without any need to resort to the introduction and tuning of additional time scales. The method is applied to the dynamical system described by the incompressible Navier–Stokes equations showing that high-order, weakly nonlinear models of bifurcating flows can be derived automatically, even for multiple codimension bifurcations. We first validate the method on the primary Hopf bifurcation of the flow past a circular cylinder and after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylinders.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 109 - 145
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Copyright
© 2015 Cambridge University Press 

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References

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), 1366–1369.Google Scholar
Balay, S., Brown, J., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H.2013 PETSc Web Page. http://www.mcs.anl.gov/petsc.Google Scholar
Carini, M., Giannetti, F. & Auteri, F. 2014a First instability and structural sensitivity of the flow past two side-by-side cylinders. J. Fluid Mech. 749, 627–648.Google Scholar
Carini, M., Giannetti, F. & Auteri, F. 2014b On the origin of the flip-flop instability of two side-by-side cylinder wakes. J. Fluid Mech. 742, 552–576.Google Scholar
Charru, F. 2011 Hydrodynamic Instabilities. Cambridge University Press.Google Scholar
Coullet, P. H. & Spiegel, E. A. 1983 Amplitude equations for systems with competing instabilities. SIAM J. Appl. Maths 43 (4), 776–821.Google Scholar
Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341–387.Google Scholar
Davis, T. A. 2004 Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2), 196–199.Google Scholar
Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P. & Iooss, G. 1987 A simple global characterization for normal forms of singular vector fields. Physica D 29, 95–127.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.Google Scholar
Haragus, M. & Iooss, G. 2011 Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems. Springer.Google Scholar
Heroux, M. A. & Willenbring, J. M.2003 Trilinos users guide. Tech. Rep. SAND2003-2952. Sandia National Laboratories.Google Scholar
Hsu, L., Min, L. J. & Favretto, L. 2001 A recursive approach to compute normal forms. J. Sound Vib. 243 (5), 909–927.Google Scholar
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, 2nd edn. Springer.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users Guide. SIAM.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.Google Scholar
Marques, F., Lopez, J. M. & Shen, J. 2002 Mode interactions in an enclosed swirling flow: a double Hopf bifurcation between azimuthal wavenumbers 0 and 2. J. Fluid Mech. 455, 263–281.Google Scholar
Marques, F., Mellibovsky, F. & Meseguer, A. 2013 Fold-pitchfork bifurcation for maps with $\mathbb{Z}_{2}$ symmetry in pipe flow. Phys. Rev. E 88 (1), 013006.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Meliga, P. & Chomaz, J.-M. 2011 An asymptotic expansion for the vortex-induced vibrations of a circular cylinder. J. Fluid Mech. 671, 137–167.Google Scholar
Meliga, P., Gallaire, F. & Chomaz, J.-M. 2012 A weakly nonlinear mechanism for mode selection in swirling jets. J. Fluid Mech. 699, 216–262.Google Scholar
Mizushima, J. & Ino, Y. 2008 Stability of flows past a pair of circular cylinders in a side-by-side arrangement. J. Fluid Mech. 595, 491–507.Google Scholar
Noak, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15–53.Google Scholar
Rehberg, I. & Ahlers, G. 1985 Experimental observation of a codimension-two bifurcation in a binary fluid mixture. Phys. Rev. Lett. 55, 500–503.Google Scholar
Roberts, A. J. 1997 Low dimensional modelling of dynamics via computer algebra. Comput. Phys. Commun. 100, 215–230.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. 1971 Nonlinear stability theory. Annu. Rev. Fluid Mech. 3, 347–370.Google Scholar
Tchoufag, J., Fabre, D. & Magnaudet, J. 2014 Global linear stability analysis of the wake and path of buoyancy-driven discs and thin cylinders. J. Fluid Mech. 740, 278–311.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Tuckerman, L. S. 2001 Thermosolutal and binary fluid convection as a 22 matrix problem. Physica D 156, 325–363.Google Scholar
Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–599.Google Scholar
Yu, P. & Yuan, Y. 2003 A matching pursuit technique for computing the simplest normal forms of vector fields. J. Symb. Comput. 35, 591–615.Google Scholar
Zhang, W. Y., Huseyin, K. & Ye, M. 2000 On the computation of the coefficients associated with high order normal forms. J. Sound Vib. 232, 525–540.Google Scholar