Centre manifold reduction and the Stuart‐Landau equation for fluid motions

Abstract

The centre manifold reduction to derive the Stuart-Landau equation is examined. A double expansion in terms of the Fourier series and linear eigenfunctions is introduced in hydrodynamic equations. A centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the Stuart-Landau equation. Through a formal expansion in linear eigenfunctions, the latter equation is shown to be equivalent with the one derived by the method of multiple scales. Numerical coefficients involved in the quintic Stuart-Landau equation are evaluated for plane Poiseuille flow, convection in a vertical slot, and Rayleigh-Bénard convection. In all the cases, the coefficients converge as a dimension of the ODE system increases and approach the numerical values obtained by the method of multiple scales.

Stability to Finite Disturbances: Energy Method and Landau’s Equation

“…Later Crouch and Herbert (1993) . The same problem was also considered by Sen and Venkateswarlu (1983) and Fujimura (1989Fujimura ( , 1991Fujimura ( , 1997) whose papers will be discussed below. Zhou (1982) developed an improved version of the classical Stuart-Watson method of 1960, assuming that both the amplitude A(t) and the angular frequency ω 1 (t) of the unstable wave disturbance vary with time.…”

“…(4.41b), another center manifold reduction scheme (called by him "the reduction scheme of the second category"), which starts with an infinite, or finite, system of ordinary differential equations (in the cases where original equations are partial-differential, this system can be derived by means of a Galerkin projection or/and a normal-mode expansion). Such reduction scheme was used, in particular, in the above-mentioned papers by Guckenheimer and Knobloch, Cheng and Chang, and Chen et al The main objecjtive of Fujimura (1997) was to prove the equivalence of Landau's equations, as given by this reduction scheme, to those derived by the method of multiple scales. To reduce the Navier-Stokes equations to a system of ordimary differential equations, a double expansion of flow fields in Fourier series and in eigenfunctions of the linear stability theory was used.…”

“…More detailed analysis of assumptions utilized in the rigorous derivations of Eq. (4.41) was undertaken in particular by Herbert (1983b) and Fujimura (1989Fujimura ( , 1991Fujimura ( , 1997) whose papers will be discussed later in this subsection. Stuart (1960) and Watson (1960a) investigated only the temporal nonlinear development of a two-dimensional wave disturbance in a steady plane Poiseuille flow.…”

“…Later Fujimura (1991Fujimura ( , 1997) studied one more method of derivation of the complex Landau Eq. (4.41b) from the equations of fluid motion.…”

Hexagons and triangles in the Rayleigh–Bénard problem: quintic-order equations on a hexagonal lattice

“…3 For some details of the centre manifold reduction, see Fujimura (1997). For eZ0, lZ0 holds, as had been pointed out by Schlüter et al (1965).…”