Dynamical system analysis of chaotic flow around two square cylinders

We investigate the chaotic dynamics of the irregular flow over two side-by-side square cylinders at a Reynolds number of 200 and a gap ratio of 1. For this set of parameters, the flow is chaotic. More specifically, the Navier-Stokes equations are linearized around the unsteady trajectory of the system, and the tangent space is analyzed. The Lyapunov exponents (LEs) and Covariant Lyapunov Vectors (CLVs) of the flow are identified using a periodic orthonormalization algorithm. The CLVs correspond to the eigenvalues and eigenvectors of typical linear stability analysis; however, the latter is performed on a steady base flow, whereas our analysis is performed on a time-varying base flow. For unsteady flows, the analysis of a time-varying tangent space is physically more meaningful and mathematically more rigorous than performing linear stability analysis on a time-averaged base flow.

We visualize the structure of the attractor by studying the spatio-temporal evolution of the CLVs. CLVs corresponding to the larger LEs consist of small-scale structures located close to the cylinders. In contrast, CLVs corresponding to the smaller LEs appear further away in the wake and consist of larger structures. To gain further insight, we apply flow decomposition techniques, such as Proper Orthogonal Decomposition (POD) and Spectral POD, to the leading CLV. The results are compared with the leading unstable eigenvector obtained using linear stability analysis.