Bridging Lyapunov stability analysis and global linear stability analysis of the chaotic flow past two square cylinders

We investigate the connection between Lyapunov stability and Global Linear Stability Analysis (GLSA) for the flow past two side-by-side square cylinders at Re = 200 and gap ratio 1. Lyapunov analysis reveals that the flow is characterised by two distinct positive Lyapunov exponents, confirming its chaotic nature. The flow is low pass filtered in the time domain using a causal exponential filter with a varying filter width Δ. As Δ increases, the chaotic base flow approaches the time-averaged field, and its Lyapunov spectrum converges to the GLSA eigenvalue spectrum computed about the time-averaged flow.

We demonstrate also that as Δ increases, the number of positive Lyapunov exponents remains unchanged at 2. However, the two unstable covariant Lyapunov vectors (CLVs) of the unsteady base flow, associated with two distinct flow phenomena - CLV 1 with gap-jet flapping and vortex shedding, and CLV 2 with co-rotating vortex-pairing, morph into to the first two GLSA eigenvectors, which are unstable and form a complex-conjugate pair. The SPOD spectra of CLV 1 and CLV 2 of the filtered base flow peak at frequencies matching those of the first two unstable GLSA modes. Additionally, two zero Lyapunov exponents appear, and their respective neutral CLVs converge to the neutral GLSA eigenvector pair further supporting the notion that GLSA is the asymptotic limit of Lyapunov stability for large Δ.