Asymptotic Center--Manifold for the Navier--Stokes

Abstract

Center-manifold approximations for infinite-dimensional systems are treated in the context of the Navier--Stokes equations extended to include an equation for the parameter evolution. The consequences of system extension are non-trivial and are examined in detail. The extended system is reformulated via an isomorphic transformation, and the application of the center-manifold theorem to the reformulated system results in a finite set of center-manifold amplitude equations coupled with an infinite-dimensional graph equation for the stable subspace solution. General expressions for the asymptotic solution of the graph equation are then derived. The main benefit of such an approach is that the graph equation, and the subsequent asymptotic expressions are formally valid even when the system is perturbed slightly away from the bifurcation point. The derivation is then applied to two cases - the classic case of the Hopf bifurcation of the cylinder wake, and a case of flow in an open cavity which has interesting dynamical properties after bifurcation. Predictions of the angular frequencies of the reduced systems are in good agreement with those obtained for the full systems close to the bifurcation point. The Stuart-Landau equations for the two cases are also obtained. The presented methodology may easily be applied to other infinite-dimensional systems.

Figures (17)

• Figure 1: Streamwise velocity of (top) the stationary base flow at $\mathrm{Re}_{c}=46.30$ and (bottom) the parameter mode.
• Figure 2: Spectrum for the flow across a cylinder at the critical Reynolds number of $Re_{c}=46.30$. The spectrum includes the parameter mode located at the origin.
• Figure 3: Streamwise velocity of the direct eigenmode. The panels represent the (top) real part and (bottom) imaginary part of the direct critical mode.
• Figure 4: Streamwise velocity of the adjoint eigenmode. The panels represent the (top) real part and (bottom) imaginary part of the adjoint critical mode.
• Figure 5: Streamwise velocity components of the restricted resolvent solutions corresponding to the zero frequency fields (top) $\mathbb{y} _{0,0}$ and (bottom) $\mathbb{y} _{1,2}$.