Vortex shedding over a cylinder is strongly affected by the cylinder oscillation. The dynamics of the cylinder wake subjected to harmonic forced excitation in the inline direction at Re = 200 is investigated in this work. Two dominant modes of the transverse velocity field are considered to model and predict the nonlinear interaction of 2D vortex shedding. The normal form symmetries have the main role in the pattern formation. The interaction of two steady modes in the presence of O(2) × S1 symmetry is described by equivariant theory. Considering the symmetries, the amplitude equations are developed with the frequency saturation information included by the addition of complex coefficients. The reduced model is expanded up to 7th order, in order to include the spatio-temporal effects. The coefficients of the model are obtained from 2D simulations of the cylinder wake flow.
The physical significance of the inline amplitude oscillation on the wake dynamics is captured by the variation of the two linear coefficients of the low order model. Similarly to the numerical results, as the amplitude of oscillation increases, two limit cycles undergo the symmetry-breaking bifurcation leading to a quasi-periodic state. The existence of the second frequency in addition to the natural shedding frequency is manifested as the small amplitude oscillation in the quasi-periodic state. For a forcing amplitude A/D = 0.5, the quasi-periodic state undergoes a torus doubling bifurcation. The dominant frequency of the bifurcated S mode matches the lift coefficient shedding frequency at A/D = 0.5 obtained from the numerical computation. The lift coefficient signal is not absolutely periodic due to the presence of the other peaks in addition to the dominant one at St = 0.1 representing the quasi-periodic flow pattern. The modulated travelling waves bifurcated from the low order model have mode S as the basic v-velocity mode which verifies the symmetric torus-doubled transverse velocity pattern observed in CFD simulation. Thus the proposed low order model can predict, with reasonable accuracy, the bifurcation sequence of the forced cylinder wake dynamic transitions observed in the numerical computation results.