A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake

Abstract

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.