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

Abstract

In the case of a cylinder flow, Barkley (EuroPhys. Lett 75, 2006) has shown, thanks to a global mode analysis, that the mean-flow was marginally stable and that the eigenfrequencies 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 analysis valid in the vicinity of the critical Reynolds number and based on the small parameter eps = Rec^(-1)-Re^(-1) << 1. We numerically compute the complex constants lambda; and mu' which appear in the Stuart-Landau Amplitude equation: dA/dt = eps*lambda* A-eps*mu'*A*|A|^2. Here A is the scalar complex amplitude of the marginally stable global mode existing at eps > 0 and which becomes unstable for eps > 0. If one looks carefully to the non-linear interactions yielding to mu';, we have shown that 1/ the mean-flow is stable 2/ the linear dynamics of the mean-flow yields the frequency of the saturated Stuart-Landau limit cycle. We will show that this result is not general by studying the case of an open cavity.