Bifurcation analysis and frequency prediction in shear-driven cavity flow

A comprehensive study of the two-dimensional incompressible shear-driven flow in an open square cavity is carried out. Two successive bifurcations lead to two limit cycles with different frequencies and different numbers of structures which propagate along the top of the cavity and circulate in its interior. A branch of quasi-periodic states produced by secondary Hopf bifurcations transfers the stability from one limit cycle to the other. A full analysis of this scenario is obtained by means of nonlinear simulations, linear stability analysis and Floquet analysis. We characterize the temporal behaviour of the limit cycles and quasi-periodic state via Fourier transforms and their spatial behaviour via the Hilbert transform. We address the relevance of linearization about the mean flow. Although here the nonlinear frequencies are not very far from those obtained by linearization about the base flow, the difference is substantially reduced when eigenvalues are obtained instead from linearization about the mean and in addition, the corresponding growth rate is small, a combination of properties called RZIF (real zero imaginary frequency). Moreover growth rates obtained by linearization about the mean of one limit cycle are correlated with relative stability to the other limit cycle. Finally, we show that the frequencies of the successive modes are separated by a constant increment.