Global dynamics of Base- and Mean-flows: the case of the cylinder and an open cavity

In the case of a cylinder flow, Barkley (EuroPhys. Lett 75, 200 6) has shown, thanks to a global mode analysis, that the mean-flow was marginally stable and that the eigenfr equencies associated to the global modes well fit the Von-Karman Strouhal(Re) function for 46 < Re < 180. The aim of this article is to give a theoretical proof of this result. For this, we achieve a weakly non-linear analys is valid in the vicinity of the critical Reynolds number and based von the small parameter ǫ = Re c − Re ≪ 1. We numerically compute the complex constants λ andμ which appear in the Stuart-Landau Amplitude equation: dA/dt = ǫλA − ǫμ′A|A|2. HereA is the scalar complex amplitude of the marginally stable global mode exis ting at ǫ > 0 and which becomes unstable for ǫ > 0. If one looks carefully to the non-linear interactions yield ing toμ, we have shown that 1/ the mean-flow is stable 2/ the linear dynamics of the mean-flow yields the frequency of t he saturated Stuart-Landau limit cycle. We will then show that this result is not general by studying the case of an open cavity.