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Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. (English) Zbl 1172.76318
Summary: This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Kármán vortex street regime that sets in for supercritical Reynolds numbers \(Re > 46\). D. Barkley [Europhys. Lett. 75, 750–756 (2006)] has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range \(46 < Re < 180\). The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number \(Re_{c}\) based on the small parameter \(\varepsilon = Re_{c}^{-1} - Re^{-1} \ll\) 1. We compute numerically the complex constants \(\lambda \) and \(\mu'\) which appear in the Stuart-Landau amplitude equation: \(dA/dt = \varepsilon \lambda A-\varepsilon \mu ' A|A|^2\). Here \(A\) is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term \(\mu '\), we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

MSC:
76E30 Nonlinear effects in hydrodynamic stability
76E09 Stability and instability of nonparallel flows in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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