Periodic Orbits and Chaotic Sets in a Low-Dimensional Model for Shear Flows

We consider the dynamics of a low-dimensional model for turbulent shear flows. The model is based on Fourier modes and describes sinusoidal shear flow, in which fluid between two free-slip walls experiences a sinusoidal body force. The model contains nine modes, most of which have a direct hydrodynamical interpretation. We analyze the stationary states and periodic orbits for the model for two different domain sizes. Several kinds of bifurcations are identified, including saddle-node bifurcations, a period doubling cascade, and Hopf bifurcations of the periodic orbits. For both domain sizes, long-lived transient chaos appears to be associated with the presence of a large number of unstable periodic orbits. For the smaller minimal flow unit domain, it is found that a periodic solution is stable over a range of Reynolds numbers, and its bifurcations lead to the existence of a chaotic attractor. The model illustrates many phenomena observed and speculated to exist in the transition to turbulence in linearly stable shear flows.