Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake

Abstract

The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the Stuart-Landau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number Re-c approximate to 46. After having reviewed Stuart's weakly nonlinear multiple-scale expansion method, we first demonstrate the crucial importance of the choice of the critical parameter. For the wake behind a cylinder considered in this paper, choosing. is an element of(2) = Re-c(1)-Re-c(-1) instead of is an element of'(2) = Re-Re-c/Re-c(2) considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech 593 333-58) correctly identified the adequate bifurcation parameter is an element of, they have plotted their results adding an additional linearization, which amounts to using. is an element of' as approximation to is an element of'. We then illustrate the risks of calculating 'running' Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to Re approximate to 100. We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuart's multiple-scale expansion method. We then briefly review a self-consistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure.