Bifurcation and stability analysis with the role of normal form symmetries on the harmonic streamwise forced oscillation of the cylinder wake

The inline cylinder oscillation effect on the wake dynamics is captured by bifurcation analysis of this model under the variation of the two linear coefficients. As the oscillation amplitude increases, two limit cycles of the model undergo a symmetry-breaking bifurcation leading to a quasi-periodic state. For amplitude-to-diameter ratio A∕D=0.5, the quasi-periodic state undergoes a torus doubling bifurcation. The bifurcated mode S frequency matches the lift coefficient shedding frequency for A∕D=0.5, obtained from the numerical (CFD) computations. The bifurcated modulated traveling waves have mode S as the dominant v-velocity mode, which confirms the symmetric period-doubled quasi-steady v-velocity pattern observed in CFD. The stability behavior of the bifurcated solution branches is also captured in the Poincare plane. The model correctly predicts the bifurcation sequence of the wake dynamic. This knowledge opens the possibility to develop low order controllers for the wake-structure interaction.