Orbital cluster-based network modelling

We propose a novel reduced-order framework to describe complex multi-frequency fluid dynamics from time-resolved snapshot data. The starting point is the Cluster-based Network Model (CNM), valued for its fully automatable development and human interpretability. Our key innovation is to model the transitions from cluster to cluster much more accurately by replacing snapshot states with short-term trajectories (“orbits”) over multiple clusters, thus avoiding non-physical diffusion of the probability distributions in the dynamics reconstruction. The proposed orbital CNM (oCNM) employs functional clustering to coarse-grain the short-term trajectories. Specifically, different filtering techniques, resulting in different temporal basis expansions, demonstrate the versatility and capability of the oCNM to adapt to diverse flow phenomena. The oCNM is illustrated on the Stuart-Landau oscillator and its post-transient solution with time-varying parameters to test its ability to capture the amplitude selection mechanism and multi-frequency behaviours. Then, the oCNM is applied to the fluidic pinball across varying flow regimes at different Reynolds numbers, including the periodic, quasi-periodic, and chaotic dynamics. This orbital-focused perspective enhances the understanding of complex temporal behaviours by incorporating high-frequency behaviour into the kinematics of short-time trajectories while modelling the dynamics of the lower frequencies. In analogy to Spectral Proper Orthogonal Decomposition, which marked the transition from spatial-only modes to spatio-temporal ones, this work advances from analysing temporal local states to examining piecewise short-term trajectories or orbits. By merging advanced analytical methods, such as the functional representation of short-time trajectories with CNM, this study paves the way for new approaches to dissect the complex dynamics characterising turbulent systems.