scholarly journals Centre-manifold reduction of bifurcating flows

2015 ◽  
Vol 767 ◽  
pp. 109-145 ◽  
Author(s):  
M. Carini ◽  
F. Auteri ◽  
F. Giannetti
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Arbitrary Dimension ◽  
Multiple Scale ◽  
Circular Cylinders ◽  
Centre Manifold ◽  
Additional Time ◽  
Flow Past ◽  
Centre Manifold Reduction ◽  
Manifold Reduction

AbstractIn 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.

Steady approach of unsteady low-Reynolds-number flow past two rotating circular cylinders

2013 ◽  
Vol 736 ◽  
pp. 414-443 ◽  
Author(s):  
Y. Ueda ◽  
T. Kida ◽  
M. Iguchi
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Vortex Method ◽  
The Self ◽  
Circular Cylinders ◽  
Far Field ◽  
Flow Past ◽  
Low Reynolds

AbstractThe long-time viscous flow about two identical rotating circular cylinders in a side-by-side arrangement is investigated using an adaptive numerical scheme based on the vortex method. The Stokes solution of the steady flow about the two-cylinder cluster produces a uniform stream in the far field, which is the so-called Jeffery’s paradox. The present work first addresses the validation of the vortex method for a low-Reynolds-number computation. The unsteady flow past an abruptly started purely rotating circular cylinder is therefore computed and compared with an exact solution to the Navier–Stokes equations. The steady state is then found to be obtained for $t\gg 1$ with ${\mathit{Re}}_{\omega } {r}^{2} \ll t$, where the characteristic length and velocity are respectively normalized with the radius ${a}_{1} $ of the circular cylinder and the circumferential velocity ${\Omega }_{1} {a}_{1} $. Then, the influence of the Reynolds number ${\mathit{Re}}_{\omega } = { a}_{1}^{2} {\Omega }_{1} / \nu $ about the two-cylinder cluster is investigated in the range $0. 125\leqslant {\mathit{Re}}_{\omega } \leqslant 40$. The convection influence forms a pair of circulations (called self-induced closed streamlines) ahead of the cylinders to alter the symmetry of the streamline whereas the low-Reynolds-number computation (${\mathit{Re}}_{\omega } = 0. 125$) reaches the steady regime in a proper inner domain. The self-induced closed streamline is formed at far field due to the boundary condition being zero at infinity. When the two-cylinder cluster is immersed in a uniform flow, which is equivalent to Jeffery’s solution, the streamline behaves like excellent Jeffery’s flow at ${\mathit{Re}}_{\omega } = 1. 25$ (although the drag force is almost zero). On the other hand, the influence of the gap spacing between the cylinders is also investigated and it is shown that there are two kinds of flow regimes including Jeffery’s flow. At a proper distance from the cylinders, the self-induced far-field velocity, which is almost equivalent to Jeffery’s solution, is successfully observed in a two-cylinder arrangement.

scholarly journals Grain boundaries in the Swift–Hohenberg equation

2012 ◽  
Vol 23 (6) ◽  
pp. 737-759 ◽  
Author(s):  
MARIANA HARAGUS ◽  
ARND SCHEEL
Keyword(s):  
Grain Boundaries ◽  
Order Approximation ◽  
Spatial Dynamics ◽  
Heteroclinic Orbits ◽  
Existence Problem ◽  
Leading Order ◽  
Centre Manifold ◽  
Centre Manifold Reduction ◽  
Manifold Reduction ◽  
Wavenumber Selection

We study the existence of grain boundaries in the Swift–Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ordinary differential equations in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

Flow past a row of flat plates at large Reynolds numbers

1993 ◽  
Vol 441 (1912) ◽  
pp. 211-235 ◽  
Keyword(s):  
Stokes Equations ◽  
Solution Procedure ◽  
Circular Cylinders ◽  
Navier Stokes ◽  
Flat Plates ◽  
Navier Stokes Equations ◽  
Element Discretization ◽  
Flow Past ◽  
High Reynolds ◽  
Outflow Boundary

The steady, incompressible, high Reynolds number, viscous flow past a row of flat plates is computed by a Galerkin finite element discretization of the Navier-Stokes equations in the streamfunction/vorticity formulation. A novel implementation of the inflow and outflow boundary conditions is described, which combines numerical stability with computational economy in the solution procedure. The calculations reported here cover the range of medium and small blockage ratios, i. e. 5 ≼ a ≼ 25 (where a is the inverse blockage ratio). A transition is found from narrow wake eddies for small values of a , to wide wake eddies for values of a above a crit ≈ 15. This transition is in general agreement with the trends reported earlier by Fornberg (1991), for the related problem of flow past a row of circular cylinders (for which a crit was approximately 8).

scholarly journals Periodic waves of the Lugiato–Lefever equation at the onset of Turing instability

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170188 ◽  
Author(s):  
Lucie Delcey ◽  
Mariana Haraguss
Keyword(s):  
Orbital Stability ◽  
Stability Result ◽  
Turing Instability ◽  
Theme Issue ◽  
Centre Manifold ◽  
Periodic Perturbations ◽  
Centre Manifold Reduction ◽  
The Stability ◽  
Bounded Perturbations ◽  
Manifold Reduction

We study the existence and the stability of periodic steady waves for a nonlinear model, the Lugiato–Lefever equation, arising in optics. Starting from a detailed description of the stability properties of constant solutions, we then focus on the periodic steady waves which bifurcate at the onset of Turing instability. Using a centre manifold reduction, we analyse these Turing bifurcations, and prove the existence of periodic steady waves. This approach also allows us to conclude on the nonlinear orbital stability of these waves for co-periodic perturbations, i.e. for periodic perturbations which have the same period as the wave. This stability result is completed by a spectral stability result for general bounded perturbations. In particular, this spectral analysis shows that instabilities are always due to co-periodic perturbations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

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