Fully nonlinear global modes in slowly varying flows

We study the existence of nonlinear solutions of the real Ginzburg–Landau amplitude equation, with varying coefficients when the solution is subject to a boundary condition at x=0. These solutions, called nonlinear global modes, are explicitly obtained from a matched asymptotic expansion when nonlinear effect dominates over the inhomogeneity. The dynamics of this model is believed to mimic the behavior of strongly nonlinear but weakly nonparallel basic flow (basic flow varying in the streamwise direction). For the model, we derive scaling laws for the amplitude of nonlinear global modes and for the position of the maximum that explain for the first time the experimental observations of Goujon-Durand et al. [Phys. Rev. E 50, 308 (1994)] and the numerical simulations of Zielinska and Wesfreid [Phys. Fluids 7, 1418 (1995)] of the wake behind bluff bodies.