A NUMERICAL AND THEORETICAL-STUDY OF THE 1ST HOPF-BIFURCATION IN A CYLINDER WAKE

The first Hopf bifurcation of the infinite cylinder wake is analysed t heoretically and by direct simulation. It is shown that a decompositio n 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 identif ied: the lock-in of frequencies in the flow and separation of the rapi d timescale of the periodicity from the slow timescale of the non-peri odic behaviour. The Landau model is investigated under weak assumption s allowing strong nonlinearities and transition to saturation of ampli tudes. It is found to be rather well satisfied locally at a fixed posi tion of the flow until saturation. It is shown, however, that no trunc ated expansion into a series of powers of amplitude can account correc tly 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 substanti ally slower than its amplification. Physically relevant characteristic s of the Hopf bifurcation under the assumption of separation of three timescales - those of the periodicity, amplification and deformation o f the mode - are suggested.