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**Three-dimensional dynamics and transition to turbulence in the wake of bluff objects.**
*(English)*
Zbl 0754.76043

A 3-D wake flow behind a circular bluff body was investigated by a numerical simulation of the Navier-Stokes-equations with a high-order time accurate mixed spectral/spectral element method. The authors reported that at \(\text{Re}=175\) the introduced 3-D noise will die out and the flow will return to its 2-D state in their numerical experiments, but at \(\text{Re}=225\) the noise will be amplified in time and evolves into 3-D coherent structures. The computational results also show that the time trace of the \(w\)-component has a much stronger modulated form than that of the \(u\)- and \(v\)-components, which exhibit weaker modulation. Moreover, at \(\text{Re}=300\), the bifurcation in the wake flow occurs, which implies that the flow leads to a periodic state with an oscillation period twice the fundamental one exhibited at \(Re=225\). The occurrence of subharmonics of the fundamental frequency in the power- spectrum of the velocity signal means that there is a period-doubling bifurcation in the phase space. So, the key finding of this paper is that the route to chaos in a 3-D wake flow behind a stationary circular cylinder is the Feigenbaum scenario which has not been reported by others in the past. The paper also noted that the period-doubling bifurcation in the wake flow does not lead to merging of the vortex in the wake flow, but however still retains its basic coherent structures. The authors also found that the wake flow at \(\text{Re}=333\) is transitional, which was computed from the velocity field at \(\text{Re}=300\). When the Reynolds number is subsequently increased to \(\text{Re}=500\), then the simulation experiment reaches a turbulent field. But, the Reynolds number Re=300 of the transition to chaos by numerical simulation seems somehow to be larger than that of the present available experimental data, \(\text{Re}\simeq 185\). In addition, the Feigenbaum route from laminar state to chaos is only observed by experiments in the wake flow behind an oscillating circular cylinder. The cause of the difference between the computational results and the experimental data is still unknown.

Reviewer: Wei Zhong-Lei (Beijing)

### MSC:

76F10 | Shear flows and turbulence |

76F20 | Dynamical systems approach to turbulence |

76E99 | Hydrodynamic stability |

76D25 | Wakes and jets |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

### Keywords:

mixed spectral/spectral element method; coherent structures; bifurcation; phase space; chaos; Feigenbaum route
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\textit{G. E. Karniadakis} and \textit{G. S. Triantafyllou}, J. Fluid Mech. 238, 1--30 (1992; Zbl 0754.76043)

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