Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder

Abstract

The aim of this paper is to provide a complete description of the bifurcation scenario of a uniform flow past a rotating circular cylinder up to Re = 200. Linear stability theory is used to depict the neutral curves and analyse the arising unstable global modes. Three codimension-two bifurcation points are identified, namely a Takens-Bogdanov, a cusp and generalised Hopf, which are closely related to qualitative changes in orbit dynamics. The occurrence of the cusp and Takens-Bogdanov bifurcations for very close parameters (corresponding to an imperfect codimension-three bifurcation) is shown to be responsible for the existence of multiple steady states, as already observed in previous studies. Two bistability regions are identified, the first with two stable fixed points and the second with a fixed point and a cycle. The presence of homoclinic and heteroclinic orbits, which are classical in the presence of Takens-Bogdanov bifurcations, is confirmed by direct numerical simulations. Finally, a weakly nonlinear analysis is performed in the neighbourhood of the generalised Hopf, showing that above this point the Hopf bifurcation is subcritical, leading to a third range of bistability characterised by both a stable fixed point and a stable cycle.