Cluster-based hierarchical network model of the fluidic pinball - cartographing transient and post-transient, multi-frequency, multi-attractor behaviour

We propose a self-supervised cluster-based hierarchical reduced-order modelling methodology to model and analyse the complex dynamics arising from a sequence of bifurcations for a two-dimensional incompressible flow of the fluidic pinball. The hierarchy is guided by a triple decomposition separating a slowly varying base flow, dominant shedding and secondary flow structures. All these flow components are kinematically resolved by a hierarchy of clusters. The transition dynamics between these clusters is described by a directed network, called the cluster-based hierarchical network model (HiCNM). Three consecutive Reynolds number regimes for different dynamics are considered: (i) periodic shedding at Re = 80, (ii) quasi-periodic shedding at Re = 105 and (iii) chaotic shedding at Re = 130, involving three unstable fixed points, three limit cycles, two quasi-periodic attractors and a chaotic attractor. The HiCNM enables identification of the dynamics between multiple invariant sets in a self-supervised manner. Both the global trends and the local structures during the transition are well resolved by a moderate number of hierarchical clusters. The proposed HiCNM provides a visual representation of transient and post-transient, multi-frequency, multi-attractor behaviour and may automate the identification and analysis of complex dynamics with multiple scales and multiple invariant sets.