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. 776821) 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 NavierStokes 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 cylinderand after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylinders
Centre-Manifold Reduction of Bifurcating Flows
CARINI, MARCO;AUTERI, FRANCO;
2015-01-01
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. 776821) 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 NavierStokes 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 cylinderand after we illustrate its application to a codimension-two bifurcation arising in the flow past two side-by-side circular cylindersFile | Dimensione | Formato | |
---|---|---|---|
CARIM_OA_01-15.pdf
Open Access dal 14/08/2016
Descrizione: Paper Open Access
:
Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione
2.45 MB
Formato
Adobe PDF
|
2.45 MB | Adobe PDF | Visualizza/Apri |
CARIM01-15.pdf
Accesso riservato
Descrizione: Paper
:
Publisher’s version
Dimensione
3.95 MB
Formato
Adobe PDF
|
3.95 MB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.