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

The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the StuartLandau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number Rec ≈ 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 ϵ² = Re- ^{- 1}-Re_c ^{- 1} instead of ϵ ′2= Re Re Re 2 c c 2 considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech 593 33358) correctly identified the adequate bifurcation parameter , they have plotted their results adding an additional linearization, which amounts to using ϵ′ as approximation to . 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 ≈ 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.