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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: [email protected]
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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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