Weak three dimensionality of a flow around a slender cylinder: the Ginzburg-Landau equation

摘要

In this paper a weak three-dimensionality of the flow around a slender cylinder is considered and the related model, the so-called Ginzburg-Landau equation, is here obtained as an asymptotic solution of the 3D (discrete) Navier-Stokes equation. The derivation is in line with existing slender bodies theories, as the Lifting Line Theory, for example, where the basic 2D flow, leading to Landau's equation, is influenced now by a "sidewash" that modifies bi-dimensionally the original flow through mass conservation. The theory is asymptotically consistent and rests on an assumption that holds in the vicinity of the Hopf bifurcation (Recr ? 45); furthermore, it leads to a well-established way to determine numerically both the Landau's coefficient μ and Ginzburg's coefficient g . Arguments are given suggesting that this assumption should hold far beyond Hopf bifurcation (Re Recr) and, with it, to extend the Ginzburg-Landau equation almost to the border of the transition region Re ? 105. In this work only the theoretical development is addressed; numerical results will be presented in a forthcoming paper.